Without directly answering the question about Drinfeld's paper, I'd emphasize the variations in the definition of "quantum group" in the literature and the resulting variation in discussions about centers, representations, etc. The early work by Drinfeld and Jimbo was partly motivated by mathematical physics, so the $q$ in the definition might be a complex number (say related to the Planck constant). But in later work by Lusztig and others the $q$ is at first an indeterminate and can then be specialized, so that $q=1$ recovers something close to the universal enveloping algebra of a complex semisimple Lie algebra whereas $q \neq 1$ a root of unity leads elsewhere. Jantzen's book mostly follows Lusztig's lead in the treatment of quantized universal enveloping algebras, where a version of Kostant's $\mathbb{Z}$-form leads to finite dimensional versions, etc. (Other generalizations involve affine Lie algebras, while very different "quantum groups" arise from function algebras of algebraic groups.)
In all of these settings it is a problem to describe the center of the quantum group and relate it to the (initially finite dimensional) representation theory.
As far as I know, only partial information about centers has been developed so far, while at least for Lusztig's quantum groups the finite dimensional representations have mostly been studied without full knowledge of the center.
That has certainly been true in modular representation theory as well.
The good classical prototype occurs in the older work of Chevalley and Harish-Chandra on the usual universal enveloping algebra of a complex semisimple Lie algebra (say of rank $\ell$). Here the center turns out to be just a polynomial algebra in $\ell$ indeterminates, closely related to the enveloping algebra of a Cartan subalgebra and characterized as a suitable algebra of Weyl group invariants. In turn, the "central characters" are easily described and help to sort out representations (as in the earlier use of isolated Casimir elements)
even in the more complicated infinite dimensional setting of the BGG category.
The notion of "infinitesimal character" even plays a role in the study of Lie group representations, but is only one of many ingredients there. Note that trace functions arising from representations are (as in Drinfeld's construction) a natural way to relate the center to the representation category.
In prime characteristic things get more complicated, as in Lusztig's work on quantum groups at a root of unity: here the finite Weyl group tends to give way to an infinite Coxeter group such as the affine Weyl group (of Langlands dual type) and much is still not understood even though the quantum group representations have been fairly well sorted out. Along the way the role of the center gets diminished, though is still potentially quite interesting.
If the question here is limited just to the Lusztig version of quantized enveloping algebras in characteristic 0 (for an indeterminate $q$), I'm not sure the center has yet been understood well enough from Drinfeld's viewpoint to contribute much to the study of finite dimensional representations. The latter closely resemble the familiar highest weight representations of a semisimple Lie algebra and can be viewed as "quantizations".
By now there is of course a lot of literature to consult, beyond the original papers by Drinfeld and Jimbo. Jantzen gives a good introduction to Lusztig's theory, while Lusztig's many papers (and one book) go farther. But the centers of the various Hopf algebras remain mysterious, to me at least.
ADDED: Probably the most useful book to consult is the 1994 Cambridge treatise A Guide to Quantum Groups by Chari and Pressley (corrected paperback reprint in 1995), which also has an extensive bibliography. They follow some of Drinfeld's 1989/1990 paper for their general discussion in 4.2A of "almost cocommutative" Hopf algebras. Chapter 10 deals with Lusztig's formulation of the quantum enveloping algebra, with $q$ an indeterminate. Here the work of Marc Rosso brings out the classical-looking role of the center in the study of finite dimensional representations. See especially Rosso's 1990 Ann. Scient. Ecole Norm. Sup. paper (and related Bourbaki seminar talk), both available online at www.numdam.org by doing a quick search for "Rosso".
In algebraic combinatorics, there is an important concept of a "$q$-analogue". Surprisingly often when you have a counting problem with a good integer answer, you realize that it can be refined to a (finite) generating function with an equally good polynomial answer. A simple example of this is the $q$-analogue of the number of permutations, which is of course $n!$. If you define the weight of a permutation to be $q^k$, where $k$ the inversion number of the permutation, then the total weight is then the important and beautiful formula
$$1(q+1)(q^2+q+1)\cdots(q^{n-1}+\ldots+1).$$
This expression is called a $q$-factorial the factors are called $q$-integers. Interesting q-analogues usually involve $q$-integers. Note that $q$-integers are closely related to cyclotomic polynomials: Every $q$-integer is a (unique) product of cyclotomic polynomials, and every cyclotomic polynomial is a (unique) ratio of products of $q$-integers.
Gaussian binomial coefficients are among the most important $q$-analogues.
The best way to think of a quantum group $U_q(\mathfrak{g})$ is that it is an algebraic $q$-analogue of a simple Lie group $G$ or its universal enveloping algebra $U(\mathfrak{g})$. For generic values of $q$, it has exactly the same (names of) representations as $G$ that tensor in the same way, plus (depending on conventions) possibly some other representations that are less important. But, what has changed is the positions of the sub-representations and the specific representation matrices. In general, when you see expressions such as integers, binomial coefficients, and factorials in the formulas for representation matrices, you see $q$-integers, Gaussian binomial coefficients, and $q$-factorials in the quantum group version. The only asterisk to this is the preferred convention of using centered Laurent polynomials (which may have half-integer exponents) rather than standard polynomials with non-negative integer exponents.
The only non-generic values of $q$ for quantum groups are roots of unity. In this case a new and also fundamental effect appears: The representation theory acquires features shared with representations of algebraic groups in positive characteristic. They have been used to strengthen or at least clarify the representation theory of algebraic groups.
The main application of quantum groups: Topological invariants. Eventually when studying the representation theory of a Lie group, you consider tensor networks, i.e., invariant tensors combined with tensor products and contractions. Because of the extra non-commutativity of a quantum group (or any non-commutative, non-cocommutative Hopf algebra), an invariant tensor network of a quantum group needs to be embedded in $\mathbb{R}^3$ in order to be interpreted or evaluated as an algebraic expression. And then the remarkable outcome is that you obtain the Jones polynomial, when the quantum group is $U_q(\text{sl}(2))$, and its well-known generalizations for other quantum groups.
Quantum groups are the main algebraic way to understand the quantum polynomial invariants of knots and links; and quantum groups at roots of unity are the main algebraic way to understand the corresponding 3-manifold invariants. In fact, this is closely connected to why they were first defined.
In response to Semyon's question in the comments: The concept of a $q$-analogue in combinatorics has never been entirely rigorous, and if anything the construction of quantum groups has been clarifying. The rough idea is that a counting problem in combinatorics is interesting when it has a "nice" answer, which often (but not by any means always) means an efficient product formula. So then a $q$-analogue is a weighted enumeration or finite generating function in which every weight is a power of $q$, and the enumeration still has all favorable numerical properties, and $q$-integers or cyclotomic factors arise.
In the case of quantum groups, first of all they are Hopf algebras. A Hopf algebra is an algebra together with all necessary extra apparatus to define the tensor product of two representations as a representation (i.e., comultiplication) and the dual of a representation as a representation (i.e., the antipode map). A universal enveloping algebra $U(\mathfrak{g})$ is of course a Hopf algebra. In this case, any deformation of $U(\mathfrak{g})$ as a Hopf algebra is potentially interesting. There is a cohomology result that if $\mathfrak{g}$ is complex and simplex, then there is only one non-trivial deformation, and you might as well call its parameter $q$ with $q=1$ at the undeformed point. (Sometimes the logarithm of $q$ is used and is called $h$, in reference to Planck's constant.) Since this is the only deformation, it is an analogue of some kind, and it is interesting. Moreover, there is a parametrization of the deformation so that $q$-integers (or quantum integers, in centered form) and cyclotomic polynomials arise in the structure of the Hopf algebra and its representations. The analogue thus deserves to be called a $q$-analogue.
Theo's explanation illustrates this more explicitly. The quantum plane is a non-commutative algebraic space (in the sense that you can interpret it as a purely formal "Spec" of the quantum plane ring) that is associated to the non-cocommutative algebraic group version of $U_q(\text{sl}(2))$. So, then, in the ring of the quantum plane, if you just expand $(x+y)^n$, you get a $q$-analogue of the binomial coefficient theorem using Gaussian binomial coefficients. (Where the $q$-exponent of a word in $x$ and $y$ is its inversion number, just as with the $q$-enumeration of permutations.) This is one of many examples where $q$-analogues that were considered long before quantum groups appear in the theory of quantum groups.
As for knots and links: In order for $U(\mathfrak{g})$ to have a non-trivial deformation as a Hopf algebra, you have to allow comultiplication to be non-commutative, even though $U(\mathfrak{g})$ itself is cocommutative. So then if $V$ and $W$ are two representations, $V \otimes W$ and $W \otimes V$ are not isomorphic via the usual switching map $v \otimes w \mapsto w \otimes v$, because that switching map is not in the category. (Due to non-cocommutativity, it is not equivariant, i.e., not an intertwiner.) However, $V \otimes W$ and $W \otimes V$ are still isomorphic, just by an adjusted version of the switching map. (You see the same theme in the category of super vector spaces, where there is a sign correction when $v$ and $w$ both have odd degree.) However, there are two natural, in-category deformations of the switching map, not just one. It so happens that they should be interpreted as left- and right- half twists in a braid group, so that you get braid representations and ultimately knot and link invariants.
The point is that ordinary tensors (and tensor networks) live in spaces that have natural actions of symmetric groups, because you can permute indices of tensors. The whole theme of quantum group definitions is deformations, and it so happens that the symmetric group action deforms into a braid group action.
This explains topological invariants such as the Jones polynomial in a one- and two-dimensional sort of way, from braids to diagrams of knots to knots themselves. It is more satisfying to have a more intrinsically 3-dimensional definition. (Actually, what counts as intrinsically 3-dimensional is somewhat debatable, but never mind that.) This is why Witten provided a "definition" of the Jones polynomial and related invariants using Chern-Simons quantum field theory. It is not really a rigorous definition, but it is very credible as a "physics definition" or even an incomplete, but maybe one-day rigorous, mathematical definition. This leads to the basic association between Chern-Simons quantum field theory and quantum groups, that they are two ways to describe the same topological invariants.
Best Answer
There are (at least) five interesting versions of the quantum group at a root of unity.
The Kac-De Concini form: This is what you get if you just take the obvious integral form and specialize q to a root of unity (you may want to clear the denominators first, but that only affects a few small roots of unity). This is best thought of as a quantized version of jets of functions on the Poisson dual group. It's most important characteristic is that it has a very large central Hopf subalgebre (generated by the lth powers of the standard generators). In particular, its representation theory is sits over Spec of the large center, which is necessarily a group and turns out to be the Poisson dual group. It also has a small quotient Hopf algebra when you kill the large center.
The main sources for the structure of the finite dimensional representations are papers by subsets of Kac-DeConcini-Procesi (the structure of the representations depends on the symplectic leaf in G*, in particular there are "generic" ones coming from the big cell) as well as some more recent work by Kremnitzer (proving some stronger results about the dimensions of the non-generic representations) and by DeConcini-Procesi-Reshetikhin-Rosso (giving the tensor product rules for generic reps). The main application that I know of this integral form is to invariants of knots together with a hyperbolic structure on the compliment and to invariants of hyperbolic 3-manifolds due to Kashaev, Baseilhac-Bennedetti, and Kashaev-Reshetikhin. The hope is that these invariants will shed some light on the volume conjecture.
The Lusztig form: Here you start with the integral form that has divided powers. Structurally this has a small subalgebra generated by the usual generators (E_i, F_i, K_i) since E^l = 0. The quotient by this subalgebra gives the usual universal enveloping algebra via something called the quantum Frobenius map. The main representation that people look at are the "tilting modules." Tilting modules have a technical description, but the important point is that the indecomposable tilting modules are exactly the summands of the tensor products of the fundamental representations. Indecomposable tilting modules are indexed by weights in the Weyl chamber. The "linkage principle" tells you that inside the decomposition series of a given indecomposable tilting module you only need to look at the Weyl modules with highest weights given by smaller elements in a certain affine Weyl group orbit.
It is the Lusztig integral form (not specialized) that is important for categorification. The Lusztig form at a root of unity is important for relationships between quantum groups and representations of algebraic groups and for relationships to affine lie algebras. The main sources are Lusztig and HH Andersen (and his colaborators). I'm also fond of a paper of Sawin's that does a very nice job cleaning up the literature.
The Lusztig integral form is also the natural one from a quantum topology point of view. For example, if you start with the Temperley-Lieb algebra (or equivalently, tangles modulo the Kauffman bracket relations) and specialize q to a root of unity what you end up with is the planar algebra for the tilting modules for the Lusztig form at that root of unity.
The small quantum group:
This is a finite dimensional Hopf algebra, it appears as a quotient of the K-DC form (quotienting by the large central subalgebra) and as a subalgebra of the Lusztig form (generated by the standard generators). I gather that the representation theory is not very well understood. But there has been some work recently by Roman Bezrukavnikov and others. I also wrote a blog post on what the representation theory looks like here for one of the smallest examples.
The semisimplified category:
Unlike the other examples, this is not the category of representations of a Hopf algebra! (Although like all fusion categories it is the category of representations of a weak Hopf algebra.) You start with either the category of tilting modules for the Lusztig form or the category of finite dimensional representations of the small quantum group and then you "semisimplify" by killing all "negligible morphisms." A morphism is negligible if it gives you 0 no matter how you "close it off." Alternately the negligible morphisms are the kernel of a certain inner product on the Hom spaces. The resulting category is semisimple, its representation theory is a "truncated" version of the usual representation theory. In particular the only surviving representations are those in the "Weyl alcove" which is like the Weyl chamber except its been cut off by a line perpendicular to l times a certain fundamental weight (see Sawin's paper for the correct line which depends subtly on the kind of root of unity).
This example is the main source of modular categories and of interesting fusion categories. Its main application is the 3-manifold invariants of Reshetikhin-Turaev (where this quotient first appears, I think) and Turaev-Viro. For those invariants its very important that your braided tensor category only have finitely many different simple objects.
The half-divided powers integral form:
This appears in the work of Habiro on universal versions of the Reshetikhin-Turaev invariants and on integrality results concerning these invariants. This integral form looks like the Lusztig form on the upper Borel and like the K-DC form on the lower Borel. The key advantage is that in the construction of the R-matrix via the Drinfeld double you should be looking at something like U_q(B+) \otimes U_q(B+)* and it turns out that the dual of the Borel without divided powers is the Borel with divided powers and vice-versa. There's been very little work done on this case beyond the work of Habiro.