- GIF (.gif): usually “line art”, drawings, etc., and not photos (though photos often come into play in “animated gifs”)
- JPG (.jpg): photos and complex art with effects or lots of unique colors/shades
- PNG (.png): will handle any image style with aplomb, but this can result in larger file sizes than if you chose the applicable one of the first two.
You probably don’t need to (and therefore shouldn’t) use TIF (.tif or .tiff) unless you are making art of some sort or doing design work. Don’t use it on the web; it probably won’t work for anybody who’s trying to view it if you do. (Likewise, don’t send people a TIF unless you like gambling on whether they will see it or be frustrated and disappointed instead.)
There you have it. That wasn’t so bad, right?