To understand what a bitcoin wallet is, you first must know what public and private keys are and how they work.
Bitcoin transactions are recorded on a public ledger known as blockchain. To receive or send bitcoins, you must have an identifier on that public ledger. The identifier is a string of 34 characters-random digits and letters. We call it a public key or a public address.
The bitcoin network subtracts the bitcoins you send from what is assigned to your public address and adds it to that of the receiver.
Every public key has a corresponding private key. Only the owner of a public key should know its private key. That is because anyone with the private key can send bitcoins from the matching public address.
Think of the public key as an email address and private key as the password to the email address.
What is a bitcoin wallet?
A bitcoin wallet is the application that holds your private keys. It is downloadable to a desktop or a mobile device. Bitcoin.org, the official bitcoin website, has a collection of wallets you can download to both your computer and mobile phone.
In addition to the software, there are hardware wallets, devices designed specifically for storing bitcoin private keys. Popular brands of hardware wallets include KeepKey, Trezo and Ledger.
Since private keys are a string of characters, you could print your wallet on a piece of paper. The data on a paper wallet representable in a QR code format, which makes it possible to scan with mobile devices. You can generate your paper wallet on Bitaddress.org.
Hot and cold wallets
Bitcoin wallets can be categorized as either cold or hot. Cold wallets are those that are stored on devices that never come into contact with the internet. It could be a wallet on an offline desktop, a hardware wallet or a paper wallet.
Hot wallets are those that are on devices that come into contact with the internet. They include wallets on desktop connected to the internet, mobile wallets and browser wallet (accessible through a browser).
Cold wallets offer better security as they are less vulnerable to remote hacks. Their physical security is however critical. If someone gets hold of your paper wallet, hardware wallet, or gets physical access to your offline desktop they can still steal your bitcoins.
When choosing a wallet, consider security and convenience. For instance, hot wallets are less secure but more convenient than cold wallets. It is advisable to keep huge sums of bitcoins in a cold wallet for long term storage. Meanwhile, you can keep bitcoin to spend on a day to day basis in a hot wallet.
You can have as many wallets as you want. There is no limit. As a matter of fact, you are supposed to use a private key only once and discard it. That way you safeguard your privacy as well as the privacy of those you transact with.