A mnemonic phrase, mnemonic recovery phrase or mnemonic seed is a list of words which store all the information needed to recover a bitcoin wallet. A wallet will typically generate a mnemonic backup 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 again and use the paper backup to get their bitcoins back.
Anybody who discovers the phrase can spend the bitcoins. Keeping the seed secret is required for keeping possession of your bitcoins.
Mnemonic phrases are a superior way of backing up and storing bitcoin so all good wallets use them.
An example of a mnemonic phrase is:
witch collapse practice feed shame open despair creek road again ice least
The order is important.
A simplified explanation of how mnemonic phrases work is that the wallet software has a wordlist taken from a dictionary, with each word assigned to a number. The mnemonic phrase can be converted to a number which is used as the seed to a deterministic wallet that generates all the key pairs used in the wallet.
The English-language wordlist for the BIP39 standard has 2048 words, if the phrase has 12 words then the number of possible combinations is 2048^12 = 2^132, so this phrase would have 132 bits of security.
It is not safe to invent your own mnemonic phrase because humans are bad at generating randomness, the best way is to allow the wallet software to generate the phrase which you write down.
Two-Factor Mnemonic Phrases
Many wallets support for the creation of a two-factor mnemonic seed. This works by the wallet creating a mnemonic phrase and asking the user for a password or extension word. Then both the mnemonic phrase and extra word are required to recover the wallet.
The password could be written down alongside the mnemonic phrase, or it could be memorized to create a two-factor mnemonic phrase where both "something you have" plus "something you know" is required to unlock the bitcoins. This feature also provides plausible deniability, because every password generates a valid seed (and thus a deterministic wallet) but only the correct one will make the desired wallet available. Or you could create two wallets which have the same mnemonic phrase but different passwords, if coerced to give up the password you can reveal the first password only and keep the second a secret.
The BIP39 standard defines a way of passphrase-protecting a mnemonic seed. A similar scheme is also used in the Electrum standard. If a passphrase is not present, an empty string "" is used instead. In the BIP39 standard the passphrase is made intentionally hard to bruteforce, PBKDF2 is a slow function to calculate and each attempt to guess a passphrase requires a few slow EC point multiplications in order to calculate bitcoin addresses and check whether there are any bitcoins held in them. Needless to say the passphrase should be long one to make it even harder to guess or bruteforce. Electrum and some other wallets call the passphrase a "seed extension" or "mnemonic extension".
Warning: Forgetting this passphrase will result in the bitcoin wallet and any contained money being lost. Do not overestimate your ability to remember passphrases especially when you may not use it very often.
Storing Mnemonic Phrases for the Long Term
Most people write down phrases on paper but they can be stored in many other ways such as memorizing, engraving on metal, writing in the margins of a book, chiseling into a stone tablet or any other creative and inventive way.
Some people get the idea to split up their phrases. Storing 6 words in one location and the other 6 words in another location. This is a bad idea and should not be done, because if one set of 6 words is discovered then it becomes easier to bruteforce the rest of the phrase. Storing bitcoins in multiple locations like this should be done via multisignature wallets instead.
Another bad idea is to add random decoy words that are somehow meaningful to you, and later remove them to be left only with the 12 word phrase. The phrase words come from a known dictionary (see next section), so anybody can use that dictionary to weed out the decoy words.
Generally a mnemonic phrase only works with the same wallet software that created it. If storing for a long period of time it's a good idea to write the name of the wallet too.
The BIP39 English word list has each word being uniquely identified by the first four letters, which can be useful when space to write them is scarce.