I like Dr. Roberts (Stanford) definition, so I'll just paste it:

> A "backdoor" in computing is a method of bypassing the normal method of authentication. Backdoors are usually inserted into a program or algorithm before it is distributed widely. They are often hidden in part of the design of the program or algorithm. In cryptography specifically, a backdoor would allow an intruder to access the encrypted information without having the correct credentials. The backdoor would either a) allow the intruder to guess the access key based on the context of the message or b) allow the intruder to present a skeleton key that will always grant him access.

Probably the most famous backdoor is the Dual_EC_DRBG which I believe had a weakness in the random number generator that leaked its internal state.