Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to improve the situation.
The definition of the quantum group I saw is that it is a Hopf algebra given by some explicit generators and relations. Though I have heard that at least the case of the quantum $sl_2$ was motivated by physics, this was not explicit enough. If someone defined the classical (i.e. non-quantum) Lie algebra $gl_n$ or the symmetric group $S_n$ using generators and relations rather than operators acting on a vector space or on a finite set respectively, such a definition would be equally unclear to me. The abstract approach to quantum groups as deformations of a universal enveloping algebras in the class of Hopf algebras is very useful, but still not very intuitive and explicit.
While any clarifying remarks would be appreciated, I can ask the following more specific questions. (1) In simple examples of quantum groups, such as quantum $sl_2, sl_n$, are there "natural" examples of their representations, like the standard representation of the classical $sl_n$, its dual representation and their tensor powers? (2) Are there mathematical problems which are not about quantum groups, but whose solution does require this notion?
Best Answer
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.