It depends totally on the data you want to store :)
If you expect to do any kind of searching then a database is almost always the best bet.
However for image data for example, it's usually better just to store them as files.
If data "feels" like it should be a file then it should probably be a file - trust your instinct there.
If data feels like it belongs in a table, then a database is a better bet.
Pretty simplistic but it's the best I can do without more information :-)
Databases are good for manipulation of large amounts of data. Their power is quite awesome - once you've done a little bit of database work, i.e. covering joins of tables, you'll begin to see the power they have.
Files are normally used for static html pages in small websites. In larger sites they're used as caches, for example. Archives are stored as files. Logs are often stored as files.
A file is better for simple storage rather than searching and manipulating the data. You can't dip into any point into a file - you need to often read the whole file which is an expensive task resource wise. However appending data to the end of a file for example is very efficient which is why they're good for storing logs.
Hope that helps