A procedure that transforms data between plaintext and ciphertext; a crypto algorithm. Basically a "cipher" is any method of encrypting text to conceal its readability and meaning. Its origin is the Arabic "sifr," meaning empty or zero.
The term cipher is sometimes used as a synonym for ciphertext (which means "encrypted text"), but it actually describes the method of encryption rather than the result. "Plain text" is what you have before encryption, and "ciphertext" is the encrypted result.
In addition to its cryptographic meaning, "cipher" is a term also used for someone insignificant, or a stylized combination of letters, as in an elaborate monogram.