A mnemonic phrase, mnemonic recovery phrase or mnemonic seed is a list of words which store all the information needed to recover a cryptocurrency wallet. Wallet software (like Jaxx or Exodus) or hardware wallets (such as Ledger Nano S or Trezor) will typically generate a mnemonic backup phrase and instruct the user to write it down on paper. If the user’s wallet is lost (computer breaks, currupted hard drive, cellphone stolen, hardware wallet destroyed), they can use the paper backup to get their crypto-currency back. Anybody else who discovers the phrase can steal the crypto coins, so it must be kept safe like jewels or cash. For example, it must not be typed into any website. Mnemonic phrases are an excellent way of backing up and storing crypto currency and so they are used by almost all well-regarded wallets.
You can read more about it in the Bitcoin Wiki.