Paint.NET (no, it's not ms paint)not quite as feature-rich as photoshop, but it's got just about everything necessary for most bg edits. the only thing I've ever wished it had was an alpha mask function of some sort, but that can be added in with plugins. I've used this for 100% of the images I've edited for osu.
GIMPpretty much as feature-rich as photoshop, but I have no idea how to use it since I never attempted to learn it. I only ever used this program to save all my .jpg images. Paint.NET's options reagarding .jpg compression aren't nearly as in depth (i.e. they're non-existent), so sometimes it's really necessary to have a separate program just for exporting jpgs (mostly for images that contain vibrant colors).
ScriptPNGThis isn't actually an editing program, it's just a lightweight tool for lossless (or lossy, though I wouldn't recommend it) compression of .png images. png is a lossless image format, so most of the time it's rather large and .jpg is more suitable (at least where osu is concerned). However, images with mostly solid colors and little noise tend to play pretty well with png, so if you ever want to use a png image like this then it's best to run it through ScriptPNG in order to make it as small as possible.