For the purpose of this article I'll be using the phrase "image editing" or "image manipulation" software to describe tools that are used to create, crop, paint, or otherwise edit photographs or other digital images. There are varying degrees of depth here. In the Windows world, users may be familiar with such programs as Adobe Photoshop, Macromedia Fireworks, or Jasc Paintshop Pro. In Linux, there is not a huge variety of comprable tools.
There are plenty of command line programs that work in Linux to manipulate images, but I'll focus on the most common GUI-based applications.
Probably the most popular example is The GIMP (GNU Image Manipulation Program). This software is very mature and widely used. I use this for my web development and it meets nearly all of my requirements. It can manipulate jpgs, gifs, pngs, and many other common file formats. Cropping, painting, drop-shadows, and other common tasks can all be accomplished in a similar manner to Photoshop or Paintshop Pro.
Krita is part of the KOffice suite and KDE. Krita is a much newer project than The GIMP, so there are fewer users, addons, documentation, features, etc. I have spent a limited ammount of time with Krita, but I will say that it shows a lot of promise. One of the selling points of Krita is that it will support CMYK (often a necessity for professional users); a feature lacking in The GIMP. Krita is in active development, so users can expect to see new features and improved usabilty rapidly.
Popular CLI image manipulation packages: ImageMagick, GD2