Roughly, bitcoins are generated by hashing some metadata about the state of previous transactions together with an incrementing nonce until you find a hash below a certain target value. If other users verify the hash and you were the first person to find one this round then you get some bitcoins.

As the computational power of the bitcoin swarm increases, the hash target is automatically decreased, leading to a roughly constant rate of generation.

Whomever gets the solution first wins, if there are money people in it, you need more computer power to be the first.