absolute-galois-groupnt.number-theory
Usually when people talk on the absolute Galois group Gℚ of ℚ they have in mind two elements they can describe explicitly, namely the identity and complex conjugation (clearly, everything is up to conjugation), although the cardinality of the group is uncountable.
Can you describe other elements of Gℚ?
What would it mean to understand this Galois group? You could mean several things.
You could mean trying to give the group in terms of some smallish generators and relations. This would be nice, and help to answer questions like the inverse Galois problem that Greg Muller mentioned, and having a certain family of "generating" Galois automorphisms would allow you to study questions about e.g. the representation theory in quite explicit terms. However, the Galois group is an uncountable profinite group, and so to give any short description in terms of generators and relations leads you into subtle issues about which topology you want to impose.
You could also ask for a coherent system of names for all Galois automorphisms, so that you can distinguish them and talk about them on an individual basis. One system of names comes from the dessins d'enfant that Ilya mentioned: associated to a Galois automorphism we have some associated data.
These two names satisfy some relations, called the $2$-, $3$-, and $5$-cycle relation, which are conjectured to generate all relations (at least the last time I checked), but it is difficult to know whether they actually do so. If they do, then the Galois group is the so-called Grothendieck-Teichmüller group.
The problem with this perspective is that the names aren't very explicit (and we don't expect them to be: we may need the axiom of choice to show they exist, and there are only two Galois automorphisms of $\mathbb{C}$ that are measurable functions!) and it seems to be a difficult problem to determine whether the Grothendieck-Teichmuller group really is the whole thing. (Or it was the last time I checked.)
However, the cyclotomic character is a nice, and fairly canonical, name associated for Galois automorphisms. We could try to generalize this: there are Kummer characters telling us what a Galois automorphism does to the system of real positive roots of a positive rational number number (these determine a compatible system of roots of unity, or equivalent an element of the Tate module of the roots of unity). This points out one of the main difficulties, though: we had to make choices of roots of unity to act on, and if Galois theory taught us nothing else it is that different choices of roots of an irreducible polynomial should be viewed as indistinguishable. Different choices differ by conjugation in the Galois group.
This brings us to the point JSE was making: if we take the "symmetry" point of view seriously, we should only be interested in conjugacy-invariant information about the Galois group. Assigning names to elements or giving a presentation doesn't really mesh with the core philosophy.
So this brings us to how many people here have mentioned understanding the Galois group: you understand it by how it manifests, in terms of its representations (as permutations, or on dessins, or by representations, or by its cohomology), because this is how it's most useful. Then you can study arithmetic problems by applying knowledge about this. If I have two genus $0$ curves over $\mathbb{Q}$, what information distinguishes them? If I have two lifts of the same complex elliptic curve to $\mathbb{Q}$, are they the same? How can I get information about a reduction of an abelian variety mod $p$ in terms of the Galois action on its torsion points? Et cetera.
If you want to go further in understanding this point of view, I would advise you to begin learning class field theory. It is a deep subject, it can be understood in a vast variety of ways, from the very concrete and elementary to the very abstract, and although superficially it appears to be limited to describing abelian reciprocity laws, it in fact plays a crucial role in the study of non-abelian reciprocity laws as well.
The texts:
Ireland and Rosen for basic algebraic number theory, a Galois-theoretic proof of quadratic reciprocity, and other assorted attractions.
Cox's book on primes of the form x^2 + n y^2 for an indication of what some of the content of class field theory is in elementary terms, via many wonderful examples.
Serre's Local Fields for learning the Galois theory of local fields
Cassels and Frolich for learning global class field theory
The standard book at the graduate level to learn the arithmetic of non-abelian (at least 2-dimensional) reciprocity laws is Modular forms and Fermat's Last Theorem, a textbook on the proof by Taylor and Wiles of FLT. But it is at a higher level again.
I don't think that you will find a single text on this topic at a basic level (if basic means Course in arithmetic or Ireland and Rosen), because there is not much to say beyond what you stated in your question without getting into the theory of elliptic curves and/or the theory of modular forms and/or a serious discussion of class field theory.
Also, as basic suggests, you could talk to the grad students in your town, if not at your institution, then at the other one down the Charles river, which as you probably know is currently the world centre for research on non-abelian reciprocity laws (maybe shared with Paris). Certainly there are grad courses offered on this topic there on a regular basis.
Best Answer
See the last of this extended answer. I'm going to part company with everyone else and say that you can describe other elements of $\text{Gal}(\mathbb{Q})$. In other words, I claim that you can identity a specific element of $\text{Gal}(\mathbb{Q})$ in a wide range of ways, together with an algorithm to compute the values of that element as a function on $\mathbb{Q}$. You can use either a synthetic model of $\overline{\mathbb{Q}}$, or its model as a subfield of $\mathbb{C}$. Although this is all doable, what's not so clear is whether these explicit elements are interesting.
The other two parts of the answer raise interesting issues, but they are moot for the original question.
This is not exactly the question, but it is related. To begin with, it is difficult to "explicitly" describe $\overline{\mathbb{Q}}$ except as a subfield of $\mathbb{C}$. I found a paper, Algebraic consequences of the axiom of determinacy (in English translation of the title) that establishes that $\mathbb{C}$ does not have any automorphisms other than complex conjugation in ZF plus the axiom of determinacy (AD). So you need some part of the axiom of choice (AC) for this related question.
As for the smaller field $\overline{\mathbb{Q}}$, the Wikipedia page for the fundamental theorem of algebra suggests that you might not even be able to construct it in the first place without the axiom of countable choice. (I say "suggests" because I'm not entirely sure that that is a theorem. Note that AC and AD both imply countable choice even though they are enemy axioms.) Any construction with countable choice isn't truly "explicit". On the other hand, if you allow countable choice, then I suspect that you can build $\overline{\mathbb{Q}}$ synthetically by induction rather than as a subfield of $\mathbb{C}$, and that you can build many automorphisms of it as you go along.
So the questions for logicians is whether there is a universe over ZF in which $\overline{\mathbb{Q}}$ does not exist, or a universe in which it does exist but has no automorphisms.
I got email about this from Kevin Buzzard that made me look again at the paper referenced by Wikipedia, A weak countable choice principle by Bridges, Richman, and Schuster. According to this paper, life is pretty strange without countable choice. You want to make the real numbers as the metric completion of the rationals. However, there is a difference between general Cauchy sequences and what they called "modulated" sequences, which are sequences of rationals with a promised rate of convergence. They cite a result of Ruitenberg that the modulated complex numbers are algebraically closed in ZF. Hence $\mathbb{Q}$ has an algebraic closure in ZF.
But it still seems possible that without countable choice, algebraic closures of $\mathbb{Q}$ need not be unique up to isomorphism, and that the complex analysis model of $\overline{\mathbb{Q}}$ might not have automorphisms other than complex conjugation.
A better and hopefully final technical answer: As mentioned, $\overline{\mathbb{Q}}$ exists explicitly (in just ZF) as a subfield of $\mathbb{C}$. You can also construct it synthetically as follows: Consider the monic Galois polynomials over $\mathbb{Z}$. These are the polynomials such that the Galois group acts freely transitively on the roots; equivalently the splitting field is obtained by adjoining just one root. The Galois polynomials can be written in a finite notation and enumerated. Beginning with $\mathbb{Q}$, formally adjoin a root of $p_n(x)$, the $n$th monic Galois polynomial, for each $n$ in turn. If $p_n(x)$ factors over the field constructed so far, the factors can also be expressed in a finite notation; take the first irreducible factor. The result is an explicit, synthetically constructed $\overline{\mathbb{Q}}$.
For comparison, let $\widetilde{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. Each element of it is computable: Its digits can be generated by an algorithm, even with an explicit bound on its running time. As we build $\overline{\mathbb{Q}}$, we can also build an isomorphism between $\widetilde{\mathbb{Q}}$. We can do this by sending the formal root of $p_n(x)$ to its first root in $\mathbb{C}$, using some convenient ordering on $\mathbb{C}$. Or we could just as well have used its last root, its second root if it has one, etc. Composing these many different isomorphisms between $\overline{\mathbb{Q}}$ and $\widetilde{\mathbb{Q}}$ gives you many field automorphisms.