If you are going to catch a ride on the Bitcoin train, you're going to need a Bitcoin wallet. While the word conjures up images of soft leather and a lot of card slots, this wallet is really a type of lockbox for the digital "keys" you use in making Bitcoin transactions. The private keys that you have to keep are what you use to access your bitcoin addresses. They are also what you use, in conjunction with the public keys, to digitally sign the Bitcoin transactions you participate in. Bitcoin wallets come in a several different forms - each specific to different devices, and you can also store your keys in "cold wallets" which are offline. You can even use paper storage and lock it up which keeps your keys out of the digital sightline altogether.
The most common forms of Bitcoin wallets are desktop, web, mobile, hardware and paper. Desktop wallets are created with software you have downloaded and store on your PC. The different versions available offer various features from what operating system they will run on to enhanced security to total anonymity. Some of the most common desktop wallets are the original Bitcoin software Bitcoin-Qt, MultiBit, Hive, Armory and DarkWallet.
Mobile wallets take advantage of your smartphone's capability to run apps that can keep private keys for your bitcoin addresses. They can even use the NFC, or near field communication (radio frequency signal), feature that most all newer smartphone models have today. This allows you to pull up the wallet and just tap and go with any merchants who have contactless payment point of sale units and accept Bitcoin for payments. This convenient form of Bitcoin wallet does have its drawbacks. It is a little less secure in some respects, and there has been some paranoia about them most notably with Apple which pulled a couple of major mobile wallets from their app store. This situation has been resolved and new versions released, but it is important to note that one of the limitations of mobile wallets is they cannot use the full blockchain to verify transactions due to the size and constant changes to the block chain. So mobile wallets use a smaller portion of the block chain along with some additional nodes that are trusted to make sure the right information is being utilized. Both Coinbase and Blockchain have mobile apps, along with Android's Bitcoin Wallet, and Mycelium.
Online wallets are just what they sound like - a wallet that is contained in software and connected to the internet, but it won't be what's stored on your own PC. Most of the Bitcoin exchanges where you can buy, sell and trade Bitcoin have wallets for your private key storage. Generally, you can access your online wallet from any source that can connect to the internet, but this convenience is offset by a drawback which comes at the set up of the wallet. If not done correctly, you could end up with the online organization in charge of your private key information. You want to avoid this loss of control. Coinbase, Circle, Block chain and Xapo are good examples of online wallets, and some of these online sources also offer offline cold storage for your wallet.
Hardware wallets are essentially tokens that create custom keys or encryption for your use when you need to process a transaction. This dedicated physical device can be more practical that some forms of online-provided offline cold storage which can be hard to access when you need it. Trezor and Ledger are two frontrunners for hardware wallets with several others in development but not yet in use.
A paper wallet uses two QR codes that are generated - one for the public address for receiving bitcoins and one for your private key that you use to spend or send bitcoins stored there. The biggest advantage of this type of wallet is that it keeps your keys from cyber attack.
You have to keep your wallet(s) safe - if you lose them or they become corrupted, you will lose the bitcoins stored at those private key addresses. Secure your wallet by strong password encryption, backing them up, use multi-signature keys or take them offline in cold or paper storage. How you want to work with Bitcoin will somewhat drive what kind of wallet will work for you, and the level of security you are comfortable with will also play a part. Some of the top wallets that offer good features are Blockchain, Coinbase, GreenAddress.it, Armory and Mycelium.