How does Prolog technically work? What's under the hood?

What are the names of the algorithms and concepts used in Prolog?

• Logic programming
• Depth-first, backtracking search
• Unification

See Sterling & Shapiro, The Art of Prolog (MIT Press) for the theory behind Prolog.

Probably it builds some kind of tree structure or directed object graph, and then upon queries it traveres that graph with a sophisticated algorithm. A Depth First Search maybe.

It doesn't build the graph explicitly, that wouldn't even be possible with infinite search spaces. Check out the first chapters of Russell & Norvig for the concept of state-space search. Yes, it does depth-first search with backtracking, but no, that isn't very sophisticated. It's just very convenient and programming alternative search strategies isn't terribly hard in Prolog.

understanding Prolog seems to be a great way to start, imho.

Depends on what you want to do, but knowing Prolog certainly doesn't hurt. It's a very different way of looking at programming. Knowing Prolog helped me understand functional programming very quickly.

My idea is to try to rebuild something similar and skipping the parser part completely

You mean skipping the Prolog syntax? If you happen to be familiar with Scheme or Lisp, then check out section 4.4 of Abelson & Sussman where they explain how to implement a logic programming variant of Scheme, in Scheme.

AI is a wide field, Prolog only touches symbolic AI. As for Prolog, the inner workings are too complex to explain here, but googling will give you plenty of resources. E.g. http://www.amzi.com/articles/prolog_under_the_hood.htm .

Check also Wikipedia articles to learn about the other areas of AI.

You might also want to read about the Warren Abstract Machine
typically, prolog code is translated to WAM instructions and then executed more efficiently.