AES has several different variants:


AES-128
AES-192
AES-256


But why would someone prefer use one over another?

For practical purposes, 128-bit keys are sufficient to ensure security. The larger key sizes exist mostly to satisfy some US military regulations which call for the existence of several distinct "security levels", regardless of whether breaking the lowest level is already far beyond existing technology.

The larger key sizes imply some CPU overhead (+20% for a 192-bit key, +40% for a 256-bit key: internally, the AES is a sequence of "rounds" and the AES standard says that there shall be 10, 12 or 14 rounds, for a 128-bit, 192-bit or 256-bit key, respectively). So there is some rational reason not to use a larger than necessary key.

A larger key size also resists better to large quantum computer attacks: Using Grover's algorithm, a brute-force attack on any k-bit key block cipher would only take $O(2^{k/2})$ steps, so a 256-bit key would still give 128-bit security, while a 128-bit key could be cracked in 2^64 operations, which is doable. But as far as I know, the threat of QC was an ulterior rationalization; also, it does not explain the 192-bit key size.
(And quantum computers of this size are not yet in sight for the next some years.)
