When dealing with Bitcoins you need what is called a "wallet".  This wallet is what contains your bitcoins and losing it will mean you lose your bitcoins.  Treat it like a real wallet.

There are two ways to get a wallet.  The first and easiest is to use an online/cloud based company like Coinbase.  When you register they make a wallet for you and you can now send/receive Bitcoins from that wallet.  The downside is that the company is in control of your wallet.  Think of this as like the Paypal option.

The other way to get a wallet is to make it yourself on your computer.  Bitcoin does not require a server or central authority to verify Bitcoins so you can store your own wallet with your money on your own devices!  This is like having cash / physical currency.  To start with you need to download an application to make the wallet.  There are many options to choose from so we will only list one: https://electrum.org  

You can download that for nearly any operating system.  After installing you will be presented with some options and you can just follow those, no special setup needed.  Then once you have it loaded, you can send and receive!  You can verify this by going to the "Receive" tab and look for the "Receiving address".  If you have a bunch of characters in there that is your address you can receive payments to.

Just keep in mind that when you send payments you need to be connected to the internet.