A wallet is a store of value, typically a collection of signing keys. The reference client stores these keys in a file called wallet.dat; online wallets and exchanges store client balances in a database, which is (hopefully) completely unrelated to key management.
There is also such a thing as a ``paper wallet'', in which signing keys, or material needed to derive signing keys, are printed or physically imprinted on some medium for permanent storage.
Wallets are not addresses. Wallets are not accounts. Wallets are not balances.