We all know that Photoshop is the biggie, and there are a few other well known pieces of software which will set you back quite a bit. We're ignoring these here, and concentrating on the free or cheap options for do-it-all type sofware; there are a few smaller tools for doing specific tasks as well.
Inkscape Something of a rarity, an open source and freeware vector drawing package; rescale your graphics with no loss of quality.
Paint.net Freeware image editing program; though it does require Microsoft's .NET framework 2.0.
Photobie Lightweight tool which concentrates on photo retouching, rather than a fully blown graphics package.
PixelToolbox Not a fully featured package, but excellent for creating tiling backgrounds, icons, cursors, that sort of thing.
TextureProcessor For creating organic style textures for backgrounds.
TheGIMP Free and open source graphics program. Aimed at the technically minded, installation might not be a trivial affair.
UltimatePaint Inexpensive shareware application which will be fine for most of your bitmap editing needs.