One of the trickier "what is the next number?" puzzles. The first few terms are 1, 11, 21, 1211, 111221.... your task is to find the rule and say what follows.

It's the Audioactive Sequence. Also known as Look and Say sequence.
You make the next number by describing the previous;

So the next would be 111221,
and then would be "three ones, two twos, one one", which is 312211.
Studied extensively by J. H. Conway