A seed is a piece of data which can be used to generate a Hierarchical Deterministic (HD) wallet. This is the only data required to regenerate any private and public keys in the wallet, so it is effective as a backup. Since the seed design is deterministic, a given seed will generate the exact same keys every time, and a single seed can generate a near-infinite number of public and private keys.
Seeds are often represented as Mnemonic phrases for easier storage and recollection. Seeds became a community standard thanks to BIP 32, and Mnemonic phrases became a community standard thanks to BIP 39.