Saif, excellent link. A bird's eye overview that manages to cover most topics, and provide details on specific vendor implementations.
I made three tries at writing an explanation, but this is really too big a topic. Check out the Hellerstein article (the one on the berkeley server that Saif linked to), and then ask about specifics.
It's worth noting that only a subset of "known good ideas" is implemented in any given DBMS. For example, SQLite doesn't even do hash joins, it only does nested loops (ack!!). But then, it's an easily embeddable dbms, and it does its work very well, so there's something to be said for the lack of complexity.
Learning about how a DBMS gathers statistics and how it uses them to construct query plans, as well as learning how to read the query plans in the first place, is an invaluable skill -- if you have to choose one "database internals" topic to learn, learn this. It will make a world of difference (and you will never accidentally write a Cartesian product again... ;-)).
