Recognizing a language given an automaton
In general, there is a dynamic programming algorithm for converting finite automata into regular expressions. But this one is simple enough that we can eyball it.

From the start state, we have two choices. So we can see our expression has the form | where _ represents the parts of the expression yet to be filled in.

Let's look at the top branch. There we must read two a's, then as many a's or b's as we want. So this part will have the form aa(a|b)*

For the second half, we see we must read at least one a or b. Then we can loop on a or b as many times a we want, and then we must read two a's. So we get (a|b)(a|b)*aa

Notice the patterns here. When we have more than one transition coming out of a state, that represents the "|" operation. A transiton on "a,b" is equivalent to having one transition on a and another on b.

So, our final expression is (aa(a|b)*) | ((a|b)(a|b)*aa).

Any time we have a cycle, either on a particular state, or on a path of states, we get something corresponding to *. And every time we have to read something in sequence, we have concatenation.