Cryptography is the study of hiding or encrypting information to make it accessible to only certain people. Modern Cryptography deals highly with Computer science, and Mathematics, especially Number Theory, while Classical Cryptography uses Linguistics.
A cipher is an algorithm for encrypting and decrypting information. The starting text is known as the plaintext while the end result after encryption is known as the ciphertext. The cipher usually has a secret parameter known as the key that (theoretically) allows you to be unable to decrypt the message even if the algorithm is known. For example, if the cipher is just to shift the letters of the alphabet by a certain number of characters (known as the Caesar Cipher), the key would be the number of letters that you shift by.
Classical Cryptography involves "pencil and paper" algorithms for encryption, such as the Caesar Cipher. These ciphers usually fall under two broad categories: Substitution, which replaces letters with other characters or symbols, and Transposition, which rearranges the order of the letters.