Your Bitcoin wallet is what allows you to transact with the world. It gives you ownership of Bitcoin addresses that you can use to receive coins from other users and then lets you send those coins onwards. Just like email, you can receive Bitcoins when you are offline, and all wallets are compatible with each other.
If you are new to Bitcoin, these wallets are a good place to begin.
Do you have a computer that you keep switched on all the time, that's connected to the Internet? You can help the community by simply running the original Bitcoin client on it. The original client is more resource intensive and will take a complete day to synchronize. After that your computer will contribute to the network by checking and relaying transactions.
Software wallets are installed on your computer. They give you complete control over your wallet. You are responsible for doing backups and protecting your money.
Bitcoin-Qt is the original Bitcoin client and it builds the backbone of the network. It offers the highest levels of security, privacy, and stability. However it has fewer features and it takes a lot of space and memory.
Multibit is a lightweight client that focuses on being fast and easy to use. It synchronizes with the network and is ready to use in minutes. Multibit also supports many languages. It is a good choice for non-technical users.
Armory is an advanced Bitcoin client that runs on top of Bitcoin-Qt. Expanding its features for Bitcoin power users. It offers many backup and encryption features, and it allows secure cold-storage on offline computers.
Electrum's focus is speed and simplicity, with low resource usage. It uses remote servers that handle the most complicated parts of the Bitcoin system, and it allows you to recover your wallet from a secret phrase.
Mobile wallets allow you to bring Bitcoin with you in your pocket. You can exchange coins easily and pay in physical stores by scanning a QR code or using NFC "tap to pay". Web wallets allow you to use Bitcoin anywhere with less effort to protect your wallet. However, you must choose your web wallet service with care as they host your bitcoins.
Web wallets host your Bitcoins. That means it is possible for them to lose your Bitcoins following any incident on their side. As of today, no web wallet service provide enough insurance to be used to store value like a bank.
Bitcoin Wallet is a lightweight mobile client for Android and BlackBerry OS. This client does not need to be associated with any online service to work. It is compatible with QR code scanning and NFC.