Once you’ve taken possession of any amount of Bitcoin, you will need to store it somewhere safe, just like cash or a credit card. To do this, you’ll need a Bitcoin wallet. There are 4 main types of wallet, but you’ll most likely only need two: one for spending, one for storage of the majority of your funds.
Bitcoin wallets are actually a lot safer than a physical one with your cash and cards; this can be stolen by anyone and compromised, but a properly secured Bitcoin wallet of the correct type is practically impossible to steal from.
The 4 types of wallet currently used are:
Web wallet (also called online wallet and hot wallet) – A wallet hosted on an online server, often of a third party such as an exchange or dedicated hosting service such as Blockchain.info. Easily accessible and can be used for transactions straight away.
Software wallet – A wallet stored on a specific computer. This is more secure than a web wallet in that cyber-attacks cannot compromise its contents (but attacks on that computer may still be able to). It can also be used instantly to make transactions. Opening a software wallet requires downloading the Bitcoin blockchain as a file onto the host computer, which requires around 20 gigabytes of space.
Cold storage wallet (also called offline wallet) – A wallet completely disconnected from the internet or a host device. This is where the majority of your bitcoins should be stored as they cannot be stolen unless you hand over all the security details voluntarily. Think of it as a safe and the two wallet types above as a physical wallet.
Hardware wallet – A cold storage wallet which uses additional hardware devices as an extra layer of security. These are not necessary for the majority of users; only those who require the ultimate in protection of large amounts.
The term “Bitcoin wallet” is a bit of a misnomer. Bitcoin wallets don't hold actual Bitcoins; those are essentially stored on the blockchain. Instead, Bitcoin wallets hold the private keys that give users the right to use those coins.
Each Bitcoin wallet comes with at least two keys (multi-signature wallets can have more): one public, and one private.
The private key you can think of like a password. The public key is an address to which people can in turn send you Bitcoin, and also identifies your wallet in the blockchain – the ledger of all Bitcoin transactions ever made. (Note that none of your personal information is attached to your wallet’s public key.)