Bitcoin wallets are a convenient piece of software that “holds” your bitcoins. You use a wallet to send and receive bitcoins. The wallet is a place that organizes your bitcoin addresses, and let’s you create transactions.
There are two main types of wallets, online and offline. An online wallet (for example the ones at blockchain.info or coinbase.com) have the convenience of being a managed and backed up service. The risk is that you don’t directly control all aspects of your coins and if something happens to the service itself you could lose your bitcoins (unlikely but it has happened). An offline wallet (for example the one in the bitcoin-qt client) give you all the control and backup responsibilities as well as reassurance that you control your own bitcoins. It’s really a trade off and something you need to play around with to figure out how comfortable you are with each type.
No matter what type of wallet software you use, create backups. One of the nice things about the fact that bitcoins aren’t truly physical things is that you can make copies. (Note that when you make copies you do not create new bitcoins for yourself…that’s kinda weird and cool at the same time.)
Wallet software is some of the more dynamic pieces of the Bitcoin ecosystem. There is a lot of innovation still to come with wallet software (for example allowing multi-signature transactions) with different types of user interfaces. Pay close attention to wallet software it’s the software you will use most often and there are lots of different capabilities.