I'm far from expect in this topic, but here's my attempt.


First, and that's something quite straightforward, people want to study `Gal Q` (this is how I will denote it; this common shortcut is defined as `Gal F := Gal \bar F/F`) because we like  Galois groups. For simple fields we know their Galois groups and we know how tremendously important they have been, from solving the equations (Abel et al) to doing group cohomology in class field theory.

As algebraic geometry matured, people learned that one of the deep reasons for importance of Galois group is that it's the **fundamental group** `pi_1(Spec F)` (depending on the definition, sometimes this is true only after a profinite completion, I'll omit this fine point further below). This allows us to use the whole apparatus of topology as well as the geometric intuition.

Representations of `Gal F` thus have a natural geometric meaning as some bundles (local systems) on the coverings of `Spec F`. This alone would make their study pretty important.
It's very reasonable to study a space in terms of geometric objects that live on it.

The whole Galois group is very complicated. Fortunately, since the Galois groups of `F_q` and `Q_p` (`p`-adic numbers) are known, we know lots of factorgroups of it. This is the standard topic of algebraic number theory courses.

Moreover, if we restrict ourselves to the *Abelian part of Galois group, its structure has been completely established by the **class field theory**. In other words, all one-dimensional representations of `Gal Q` are known. That begs a natural question about higher-dimensional reps — indeed this goes on by the name of **Langlands program**.

The Langlands program is such a huge topic that I don't feel able even starting to talk about it. It might be a good idea to post questions here if you'll feel brave enough to learn it :)


One more topic in the discussion of the structure of `Gal F` is about the specific (conjugacy classes) of operators that live there, called **Frobenius operators**. The amazing thing about them is that they behave, formally, in a way similar to the knot operators in the fundamental group of a threefold without knots. This image is reinforced by the computations of the dimension of `Spec Z` that give an answer of 3, providing fruitful connections to the theory of real 3-manifolds, knots and their L-functions.


Finally, here's the direction that was explored by many people and was probably made famous through Grothendieck's work. Consider a variety `Z := P^1-(0, 1, \infty)`, the projective plane without three points. It is really interesting as it makes sense over any field. Now suppose we consider it as a scheme over Q and ask for its fundamental group. Then one would have an exact sequence, if I'm not mistaken, 

    0  -->  pi_1(Z/\bar Q)  -->  pi_1(Z/Q)  -->  Gal Q --> 0

which implies, by standard reasoning, that `Gal Q` acts on `pi_1(Z/\bar Q)`, which is a very straightforward group — it's simply a topological fundamental group of a plane without two points, so it's freely generated by two loops. Grothendieck called it a *cartographic group* and related it to **dessin's d'enfants**. The standard references are very well-written in T.'s answers and there are hopefully more questions on this topic coming to MathOverflow ;)


As you see, this is a very interesting direction that we should continue to explore. One more thing I like about it is that it's also very suitable to MathOverflow format, since there are both many specific questions as well as opportunities to write reviews and to produce new results by collaboration.