Private and secret keys consists of secret information used as input to various kinds of cryptographic algorithms such as encryption, signature and MAC to select the concrete transformation done by the algorithm.
Asymmetric cryptography commonly uses pairs of one private and public keys. Symmetric cryptosystems use the same secret key for all participants.
Algorithms and protocols should be designed not to leak information about the secret/private key even if the attacker can control all other input and see all output.
Suitable questions for this tag could be about key generation, key size considerations, key agreement (DH) or key derivation. Key derivation can be split into derivation from other keys (KBKDF) or from passwords (PBKDF).
Sometimes - notably in books / lectures of Dr Katz - secret keys are used in asymmetric cryptosystems while private keys are used for symmetric cryptosystems. This is however usually not the case.
