A seed phrase, seed recovery phrase or backup seed phrase is a list of words which store all the information needed to recover Bitcoin funds on-chain. Wallet software will typically generate a seed phrase and instruct the user to write it down on paper. If the user's computer breaks or their hard drive becomes corrupted, they can download the same wallet software again and use the paper backup to get their bitcoins back.
"Always back up your recovery seed phrase by writing it down."
“Be sure to write down your recovery seed phrase.”
“Never store your seed phrase online or on any digital storage.”
“If you lose (or are locked out of) your bitcoinwallet, you can reclaim your funds with your seed phrase.”
Definition(s) from the Web
A seed phrase, seed recovery phrase or backup seed phrase is a list of words which store all the information needed to recover Bitcoin funds on-chain. Wallet software will typically generate a seed phrase and instruct the user to write it down on paper. If the user’s computer breaks or their hard drive becomes corrupted, they can download the same wallet software again and use the paper backup to get their bitcoins back. Anybody else who discovers the phrase can steal the bitcoins, so it must be kept safe like jewels or cash. For example, it must not be typed into any website. Seed phrases are an excellent way of backing up and storing bitcoins, so they are used by almost all well-regarded wallets. Seed phrases can only backups funds on the block chain. They cannot store funds involved in off-chain transactions such as Lightning Network or Blinded bearer certificates. Although these technologies are in their infancy as of 2019 so it’s possible in future seed phrases could be used to backup them. Source
When choosing a wallet for storing Satoshi, it is recommended to give preference to one that provides the ability to save private keys or seed phrase. Source
With a deterministic wallet a single key can be used to generate an entire tree of key pairs. This single key serves as the root of the tree. The generated mnemonic sentence or word seed is simply a more human-readable way of expressing the key used as the root, as it can be algorithmically converted into the root private key. Those words, in that order, will always generate exactly the same root key. A word phrase could consist of 24 words like: begin friend black earth beauty praise pride refuse horror believe relief gospel end destroy champion build better awesome. That single root key is not replacing all other private keys, but rather is being used to generate them. All the addresses still have different private keys, but they can all be restored by that single root key. The private keys to every address it has and will ever give out in the future can be recalculated given the root key. That root key, in turn, can be recalculated by feeding in the word seed. The mnemonic sentence is the backup of the wallet. If a wallet supports the same (mnemonic sentence) technique, then the backup can also be restored on another software or hardware wallet. A mnemonic sentence is considered secure. The BIP-39 standard creates a 512-bit seed from any given mnemonic. Every passphrase leads to a valid wallet. If the wallet was not previously used it will be empty. In a non-deterministic wallet, each key is randomly generated on its own accord, and they are not seeded from a common key. Therefore, any backups of the wallet must store each and every single private key used as an address, as well as a buffer of 100 or so future keys that may have already been given out as addresses but not received payments yet. Source