When viewing a question, under the question's vote count is a star button. Clicking that button marks that question as a 'favorite'.
Marking a question as a favorite has no effect on that question. You have unlimited favorites, and can undo a favorite at any time.
When you mark a question as a favorite, you can track all changes to it, even if you didn't write it. This can be done by going to the 'favorites' tab on your profile and clicking on 'activity'.
So if you reallywant to know the answer to someone else's question, or keep track of an article in general, the favorite is the way to go.
Getting 25/100 favorites on a question gives a silver/gold badge.