The advantage of using DXT compression is that they use 1/4 (for DXT3 and DXT5) or 1/8 (for DXT1) of the video memory since they are native formats for graphics cards.
The type of color compression works well for images that don't have many sharp color changes but it doesn't preserve sharp details very well. Here are some links from one of my other posts.
UnrealWiki (general overview)
Wikipedia (more technical)
MSDN (most technical)
The quick guide is:
DXT1 (called BC1 in DirectX10): 8:1 compression, use if you don't need an alpha channel or a 1 bit (on or off) alpha channel.
DXT3 (called BC2 in DirectX10): 4:1 compression, same type of color compression as DXT1 and best for sharp, hard-edged alpha transitions.
DXT5 (called BC3 in DirectX10): 4:1 compression, same type of color compression as DXT1 and best for smooth, more gradual alpha transitions.