Re: Indeed, who *does* use TIFF as a common file format?
TIFF is commonly used by scanning software, and by "imaging" (i.e., taking physical documents, scanning them, and storing and retrieving them) applications generally.
It was invented by Aldus and is now controlled by Adobe (in the sense that they hold the copyright on the specification). Derivative formats have been published by ISO and the IETF (eg RFC 2306).
One main advantage of TIFF, as Chemist noted, is that it can be used to store images in lossless encodings (uncompressed or LZW); it can also be a container for lossy-compressed JPEG images, so it's more flexible than JPEG1 or, say, PNG alone. It also supports multiple images ("pages") per file, layers, various sorts of metadata, etc.
1JPEG does define a lossless mode, but apparently it's not widely supported.