PNG - Portable Network Graphics
PNG is the new graphics format with lossless compression. It has support for most pixelformats, including alpha channel. It is also free to use.
TGA File format specification V2.0
The article written by the inventors of the specification.
Free jpg library.
Free open source image loading/saving programming library.
Free open source image saving/loading programming library.
DDS File Reference
Microsoft's file format reference for DirectDraw Surface files (.dds). Only explains the file layout, not the available compression algorithms for DXT1 through 5.
squish - DXT Compression Library
Open source library under MIT license to perform DXT compression for images. Supports DXT1, DXT3, and DXT5.
Open Source and free implementation of the JPEG 2000 image standard.
Real-time YCoCg-DXT Compression
Sep, 2007 - J.M.P. van Waveren, Ignacio Castaņo
Very good texture compression for realtime usage, e.g. in games.
Understanding BCn Texture Compression Formats
2012 - Nathan Reed
This articles gives a good overview of the commonly used hardware texture compression formats. All versions from 1 to 7 are explained. Also known as DXT compression.