What is a wallet, and what is a mnemonic / seed phrase?

Question

Accepted Answer

A wallet is software that stores your keys and lets you sign transactions; a mnemonic (seed phrase) is a list of words — often 12 — from which many private keys can be regenerated.
A wallet (such as a browser extension wallet) does not hold your money the way a bank account does — the funds live on the blockchain. What the wallet holds is your keys, and it uses them to create and sign transactions. A single mnemonic / seed phrase (a human-readable list of words, commonly 12) is the master secret: from it the wallet can deterministically derive many separate accounts, each with its own private key. The security consequence is sharp:

Leak one account's private key → an attacker controls that one account.
Leak your mnemonic → an attacker controls every account derived from it.

So both must be guarded, and the mnemonic especially — anyone who sees it can drain all your accounts. "Not your keys, not your coins" captures the stakes.
Go deeper:

Cryptocurrency wallet — Wikipedia — how wallets hold keys (not coins) and how a seed phrase deterministically regenerates them.