A Bitcoin wallet is simply a place where you can store bitcoins. A Bitcoin wallet can take many forms (from an app on your phone to a piece of paper), but it's probably easiest for beginners to think of it like a bank account for bitcoins, with one important difference: with Bitcoin, you have the option to “be your own bank”.
In other words, with Bitcoin, you don't have to trust a third party to hold your money (though you can if you wish).
Bitcoin wallets come in many forms. One of the easiest ways to get up-and-running with a wallet for those that own a smart-phone, is to simply install a Bitcoin-wallet app. One widely used option, for both iOS and Android, is BRD Wallet (previously known as “Bread” wallet). Another option you might check out for Android is Mycelium.
For additional options — for example a wallet you can use on a desktop or laptop computer — you might want to try the tool at Bitcoin.org.