Can anybody list me all the possible databases that can be used with php? Which would you use and why?
You can use all databases they support a native binding or ODBC-interface.
For beginners MySQL is very popular.
MySQL is popular cos it's easy, however you soon get annoyed with it's total and absolute lack of any sort of advanced feature.
So, I recommend PostgreSQL. Just read the docs and learn how to optimise and use it properly.
Cheers,
Chris