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Wallet.
Any individual or entity wishing to exchange Bitcoin (and not store them on an exchange, in someone else's custody) must create a digital collection of the credentials, known as a wallet, necessary to transact coins.
Full clients.
This is a wallet that includes a full copy of the entire blockchain.
This is the safest form of storage other than offline, or "cold storage," but it requires substantial digital space.
Lightweight clients.
This is a wallet that includes a more limited version of the blockchain to enable it to be portable on devices, such as a smartphone.
Since the entire blockchain is not available, a party using a lightweight wallet must trust intermediaries who have full wallets.