PCB colors - what's available, and why?

The color comes from the soldermask, so what you're really asking is what colors of soldermask are available. The answer is that you can get just about any color soldermask, but the less common ones may cost more depending on your manufacturer. Different manufacturers offer different standards. One manufacturer may only offer green as standard and you'll need to pay a premium for blue, red, etc.

To answer your main question about what is commonly available, I have seen green, blue, purple, red, black, and white. Those seem to be the most popular. At work we use green boards to indicate leaded assemblies (because in the past, prior to RoHS restrictions, all of our boards were done in green), blue to indicate RoHS-compliance, and red for prototypes. There is no "best color", it is completely up to you. Personally I don't care for black soldermask or white soldermask because it is difficult to see the traces which could be useful for troubleshooting.

As said by others, the color is the result of the solder mask used.

However, it's definitely not true that color is the only difference. For example, Paul Stoffregen of Teensy fame found that a black solder mask had reduced resolution than a green solder mask, and this directly led to lower reliability of black boards.

It's just paint so various colours are available. Okay, it's a special kind of paint so you may get worse specs on not green colour as that's by far most common one. Specs meaning the smallest bridge and clearances.

I've made black and white boards, LED backlights are white for obvious reasons. If you just take regular white colour solder resist, you're liable to end up with pink as the copper will show through. Red is also seen from time to time.

Black PCB with gold (ENIG) treating looks cool but it's harder to see the traces than on green PCB.

Usually I spec on my PCBs "Black if no extra cost"