A MAC or Message Authentication Code protects the integrity and authenticity of a message by allowing verifiers to detect changes to the message content. It requires a random key generation algo that produces a per-message random key K, a signing algorithm which takes K and message M as input and produces signature S, and a verifying algorithm with takes K, M and S as input and produces a binary decision to accept or reject the message. Unlike a digital signature a MAC typically does not provide non-repudiation. It is also called a protected checksum. Both sender and recipient of the K and M share a secret key.
HMAC: So called Hashed-MAC because it uses a cryptographic hash function, such as MD/SHA to create a MAC. The computed value is something only someone with the secret key can compute (sign) and check (verify). HMAC uses an inner key and an outer key to protect against length extension and collision attacks on simple MAC signature implementations. RFC 2104. It is a type of Nested MAC (NMAC) where both inner and outer keys are derived from the same key, in a way that keeps the derived keys independent.
HOTP: HMAC-based One Time Password. HOTP is based on an incrementing counter. The incrementing counter serves as the message M, and when run through the HMAC it produces a random set of bytes, which can be verified by the receiving party. The receiving party keeps a synchronized counter, so the message M=C does not need to be send on the wire. RFC 4226.
TOTP: Time-based One-time Password Algorithm . TOTP combines a secret key with the current timestamp using a cryptographic hash function to generate a one-time password. Because network latency and out-of-sync clocks can result in the password recipient having to try a range of possible times to authenticate against, the timestamp typically increases in 30-second intervals. Here the requirement to keep the counter synchronized is replaced with time synchronization. RFC 6328.