An important result in algebraic geometry is the Kodaira--Enriques classification. It is a description of the geography of compact complex surfaces, later extended to surfaces in characteristic $p$. The Wikipedia page contains a wealth of information (seriously, it is one of the best mathematics pages in my opinion), and provides a pretty picture that describes the classification in an easy to remember way.
Together with Johan Commelin I made an interactive version of this picture, where you can get more information about the objects that are involved: more numerical invariants, a short description and references to the literature. You can find it at http://superficie.info. Johan has written a really detailed explanation on his weblog, you should check it out!
It is far from complete, in the sense that we certainly haven't added every class of surfaces in the literature, and the existing descriptions can be improved. Neither of us is an expert in the birational classification of surfaces, but if you are and you want to contribute please get in touch with us! Also, if you want to contribute to the coding there is the GitHub project.
tl;dr: Go to superficie.info.