In order to use bitcoin, you will first need a wallet to store it in. You could think of a Bitcoin wallet as being similar to a bank account, or in some cases a service like Google Wallet or PayPal.
The two main options are: (1) a software wallet stored locally on the hard drive of your computer or smartphone, or (2) an online, web-based service.
Local and online wallets both have advantages as well as vulnerabilities. If you store bitcoin locally, make sure you back up your wallet regularly in case the hard drive crashes or becomes corrupted. Online web wallets have the advantage of being easier to access and not tied to one piece of hardware that will take your Bitcoin balance with it if it fails, but they employ varying degrees of security against hackers, from quite good (multi-factor authentication and multi-signature) to quite poor (ID and password).
There is a wide range of options when it comes to online wallets. Some are simple wallets just for storing Bitcoin, while others have Bitcoin purchasing built in, and allow you to exchange fiat currency from your bank account for bitcoin in the wallet, somewhat like transferring money to a PayPal account.