Tweak is correct.
JPG: Made to give the best quality for photographs, though can be used for any kind of static (non-animated) images. Depending on the graphics program you save it with, JPGs can look extremely crappy, or quite decent. (For example, saving a JPG in MS Paint could often leave your image looking speckled and blotchy, while saving it in Photoshop won't affect the quality noticeably.) It also offers the smallest file size, generally.
GIF: Primarily used for animated images and pixel art. For static images, this file type is designed to give better quality when you have large blocks of color (flat colored images, for example, without a ton of shading or details). GIFs can give you a transparent background, but there is no semi-transparency. That means that you won't be able to have any soft gradients turning into transparent space; it has to have a hard edge where it meets the transparent area(s).
PNG: Generally acceptable as the best overall quality with reasonable file size. Best used for static images. PNGs can give you both a transparency AND semi-transparency, so you can have soft edges unlike with GIFs.