Wallet

A wallet stores the public and private keys which can be used to receive or spend the cryptocurrency. A wallet can contain multiple public and private key pairs. The cryptocurrency itself is not in the wallet. In case of bitcoin and cryptocurrencies derived from it, the cryptocurrency is decentrally stored and maintained in a publicly available ledger.

Wallet access
When choosing a wallet, the owner must keep in mind who is supposed to have access to (a copy of) the private keys and thus has potentially access to the cryptocurrency. Just like with a bank, the user needs to trust the provider to keep the cryptocurrency safe. Trust was misplaced in the case of the Mt. Gox exchange, who 'lost' most of their clients' bitcoins. Downloading a cryptocurrency wallet from a wallet provider to a computer or phone does not automatically mean that the owner is the only one who has a copy of the private keys. For example with Coinbase, it is possible to install a wallet on a phone and to also have access to the same wallet through their website. The software can also have known or unknown vulnerabilities. For receiving cryptocurrency, access to the receiving wallet is not needed. The sending party only needs to know the destination address. Anyone can send cryptocurrency to an address. Only the one who has the private key of the corresponding address can use it.