What is a Seed phrase?
A seed phrase, seed recovery phrase or backup seed phrase is a list of words which correspond to the master private key of a Bitcoin wallet.
In typical usage, a wallet will 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 deterministically re-generate their wallet. Seed phrases are the most frequently encountered means of backing up Bitcoin wallets.
Anybody with access to the phrase can steal the bitcoins, so it must be kept safe. Phrases should not be transmitted electronically and should never be written on any website.
An example of a seed phrase is:
witch collapse practice feed shame open despair creek road again ice least
The word order is important.
Alternative name "Mnemonic Phrase"
Seed phrases are sometimes called "mnemonic phrases" especially in older literature. This is a bad name because the word mnemonic implies that the phrase should be memorized. It is less misleading to call them seed phrases.
Explaining it another way
A simplified explanation of how seed phrases work is that the wallet software has a list of words taken from a dictionary, with each word assigned to a number. The words in the seed phrase are each converted to their corresponding number and concatenated to generate the seed integer to a deterministic wallet. From this seed integer the wallet generates all key pairs used in the wallet.
The English-language wordlist for the BIP39 standard has 2048 words, so if the phrase contained only 12 random words, the number of possible combinations would be 2048^12 = 2^132 and the phrase would have 132 bits of security. However, some of the data in a BIP39 phrase is not random (see: BIP39: Generating the mnemonic), so the actual security of a 12-word BIP39 seed phrase is only 128 bits.