Image manipulation applications are used to manipulate pictures, such as removing red eye, and other more sophisticated tools.

Specific applications

ImageMagick is great for the command-line user. It also has some GUI capabilities, but is wonderful for scripting activity. With relatively simple shell scripts and ImageMagick it is possible to do basic image manipulation on hundreds of images at once without manual intervention.
Gimp is the classic Linux image manipulator. It isn't strictly image manipulation, though. There are a large variety of tutorials regarding the program.
XnView is an image viewer, with the abilities of editing and format convertion
Gqview is an image viewer that has many useful features, such as "auto rename", which makes it easy to rename a bunch of pictures at once.
Gliv is an image viewer that uses OpenGL to scale and rotate images. This means image operations are exceptionally fast.
XV
Graphviz is a collection of tools for creating directed and undirected graphs

Panoramic applications

Hugin - Front-end to Panorama Tools. Used for stitching multiple pictures together.
Enblend - Used for smoothing seams on images created from stitching tools.
PTViewer - Web based viewer to panoramic photos.

For vector image manipulation see Vector image editors