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. If you're on iOS (iPhone), Bread Wallet is an easy-to-use option. For Android, you might check out Mycelium.
For a cross-platform option — something you can install on most smart-phones and on desktop/laptop computers as well — check out Copay. Additionally, Copay has support for some advanced security options, like multi-signature wallets.