Modern cryptographic systems include Symmetric Key Cryptography algorithms (such as DES and AES) and Public Key Algorithms (such as RSA).
Symmetric Key Cryptography algorithms utilizes a Cryptographic Key
Public Key Algorithms outputs the Private Key and a corresponding Public Key.
Cryptography uses integers for Cryptographic Keys. In some cases keys are randomly generated using a Pseudorandom number generator (PRNG).
In other situations, the Key Generation is derived deterministically using a password and a Key Derivation Function (KDF).
Many Cryptography protocols are designed to have Perfect Forward Secrecy, which requires Key Generation of a new Cryptographic Key for each session.
NIST Special Publication NIST.SP.800-133 covers Recommendation for Cryptographic Key Generation