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.
A seed is simply entropy—a random string of digits. A seed is used to generate a single extended private key (xprv), which is called the master private key. This private key can be used to generate child private keys as well as public keys, allowing a wallet to generate as many key pairs as a user requires. This setup maximizes the ease of backing up a wallet with the privacy benefits of avoiding address reuse.
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.