The best way to contribute is find bugs in stuff you use and fix them, or write guides/docs for popular software. You should work on things you know and use so you understand what the users need.
Somehow I can't tell you any specific examples off the top ofbmy head.
I agree, but I want to add that a good reporting of a bug is also very helpul. (Remember to add steps to reproduce the bug, what you expected and what you got, and be nice.) And if you take a look at the fix, perhaps it's useful to fix a similar problem yourself later.
Problem with bug reports: You don't need to be a programmer to report bugs well, and the maintainer usually will fix the bug better.
I hack and track down the lines of code responsible for the issue (which allows the maintainer to quickly patch the issue), but can't confidently implement large patches as I often don't even know the libraries being used.
"Real programmers" know exactly how the codebase is structured and confidently fix both their own and other people's bugs. (Maybe this is just my lack of confidence)
Somehow I can't tell you any specific examples off the top ofbmy head.