Similar to a traditional wallet you may carry in your pocket, a bitcoin wallet is used to store money. The difference is that instead of storing a collection of bills and cards, a bitcoin wallet stores a collection of bitcoin private keys. Typically a wallet is encrypted with a password or otherwise protected from unauthorized access.
Unlike the Bitcoin network, a Bitcoin wallet is controlled only by its owner (it's not distributed and shared like the blockchain). It's important to keep your bitcoin wallet safe by either setting a strong password or otherwise keeping it out of reach of malicious individuals.
Types of Wallets
There are many different bitcoin wallets to choose from, each providing different features. All bitcoin wallets must conform to the established Bitcoin protocol so they can all be used to send and receive funds regardless of their specific implementation. This is similar to email where you have many choices of email providers but they all "speak the same language" and are able to send messages to each other.
There are several main categories of Bitcoin wallets:
Web wallets allow you to send, receive, and store bitcoin all through your web browser. They are typically hosted by a provider that manages the security of the private keys associated with your account, although some types of web wallets provide ways for you to remain in control your private keys. Web wallets are generally more convenient than other types of wallets since you don't have to worry about accidentally deleting a wallet from your computer (and losing your bitcoins).
The main criticism of most web wallets is that you relinquish some control since web wallets usually manage private keys on your behalf. This means that security must be taken very seriously by the wallet provider. Like any other online account, it's also important for customers to take some security precautions.
There are technologies available that allow users to retain full control of their funds yet take advantage of the convenience of web wallets such as Coinbase's multisig vault.
It's also possible to install wallet software directly on your computer. This allows an individual to have full control (and responsibility) over their wallet.
In a desktop wallet, the private keys are stored on a hard drive so it's only possible to access the funds using the computer the wallet is installed on. If the wallet file were to become corrupted with no backup available, the bitcoins stored in that wallet would be lost forever. For this reason it's extremely important that strong passwords are used and reliable backups are made of any desktop wallet. It's equally important that the wallet and any backups are kept safely out of reach of malicious individuals.
Desktop wallets fall into two main categories known as "full nodes" or "light" clients. Full nodes host a full copy of the blockchain (about 31 GB as of January 2015) while light clients provide only bitcoin storage capabilities while depending on an external source to read the blockchain. More information and examples of wallets can be found here.
Mobile wallets are simply bitcoin wallets designed for a mobile device. This means they can easily scan QR codes, are easy to navigate with a touch screen, and are accessible while on the move. The Coinbase mobile wallet for iOS and Android allows you to access your Coinbase account to send, receive, buy, and sell bitcoin.
Mobile wallets are almost always "light" clients in that they do not store a full copy of the blockchain.
A hardware wallet is a specialized type of device designed specifically to store bitcoins. The advantage is that hardware wallets are much more difficult for a malicious user to compromise when compared to a desktop or mobile wallet since they use the bare minimum amount of software required to safely store bitcoins.
Paper wallets are a way to embody bitcoins in a physical medium such as paper or metal. Like a printed bank note, if a paper wallet is lost or destroyed, then the bitcoin stored on it are gone forever.
A brain wallet is a Bitcoin wallet that is generated from a passphrase. It's similar to a paper wallet in that if the passphrase is lost, so are the bitcoins stored in that brain wallet. While it may be appealing to store bitcoins entirely within your memory, it should be cautioned against due to the challenge of using a sufficiently secure passphrase.