According to this book, An Introduction to the Langlands Program,
One of the fundamental goals of modern number theory is to understand the Galois group $Gal(\bar k /k)$ where $k$ is a local or a global field.
What is meant by understanding this group?
On
The answer of @BenS. is best, but another way of answering this would be to say that we can understand the full Galois group when we can describe all finite quotients (by normal closed subgroups). For $\Bbb Q$, this would mean describing the Galois groups of all normal extensions of $\Bbb Q$. As I’m sure you know, we’re not yet able to do this.
On
There is a big Mathoverflow discussion with the same title and 10 answers, some of them are quite informative depending on reader's background. It begins just like this question,
I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G=$Gal$(\overline{\mathbb{Q}}/ℚ)$. What do people mean when they say this?
One possible interpretation is that we would like to be able to write down an explicit name for the group; that is, we would like to identify what group it actually is, and what the properties of that group are: what does its center look like, what are the irreducible representations, etc.
Frequently, what is meant is that we would like to understand the representation theory of the group. If we can compute the characters of the irreducible representations, for example, we can hope to discover information about associated functions, such as $L$-functions, theta functions, modular functions, and more.