Ste's right. The 32 means colors are encoded using 32 bits (4 bytes). This means the computer can recognize 2^32 = 4,294,967,296 different colors. At 24 bits, it's 16,777,216 colors ("16 millions colors") and at 16 bits, you get 65,536 colors. The old "256 colors" corresponds of course to 8 bits. The higher the "bit count", the smoother the colors will be.
Edit: For those interested, Wikipedia showed me wrong.
"32-bit color" is a misnomer when regarding display color depth. A common misconception is that 32-bit color produces 4,294,967,296 distinct colors. In reality, 32-bit color actually refers to 24-bit color (Truecolor) with an additional 8 bits either as empty padding space or to represent an alpha channel.