A cryptocurrency 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.There are over nine hundred cryptocurrencies; the first and best known is bitcoin. Some wallets support multiple cryptocurrencies. 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. Every piece of cryptocurrency has a private key. With the private key, it is possible to write in the public ledger, effectively spending the associated cryptocurrency.
When choosing a wallet, you must keep in mind who has access to (a copy) of your private keys and thus has potentially access to your ‘money’. Just like with a bank, you have to trust the provider to keep your cryptocurrency safe. Trust was misplaced in the case of the Mt. Gox exchange, who ‘lost’ most of their clients bitcoins. Downloading a cryptocurrency wallet form a wallet provider to your computer or phone does not automatically mean that you are the only one who has a copy of your private keys. For example with Coinbase you can install a wallet on your phone, but you can also access the same wallet through their website. The software can also have consciously or unconsciously vulnerabilities. For receiving cryptocurrency you do not need access to your wallet. 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.
A backup of a wallet can come in different forms like:
A (encrypted) file like wallet.dat or wallet.bin which contains all the private keys.
A mnemonic sentence from which the root key can be generated, from which all your private keys can be recreated. Preferably these words could be remembered or written down and stored on a physical other location(s).
A private key like: KxSRZnttMtVhe17SX5FhPqWpKAEgMT9T3R6Eferj3sx5frM6obqA.
When you lose your private keys and your backup then you lose your cryptocurrency! There is no backdoor. When using a webwallet, the private keys are managed by the provider. When owning cryptocurrency you should carefully select who you trust managing your cryptocurrency. An (encrypted) copy of your wallet should be kept in a trusted place. Preferably off-line. Some people ‘write’ their mnemonic sentence or private key on metal.