Bitcoins are entries in the shared public ledger called the blockchain. When you receive bitcoins, you get a private key, like a password, that allows you to transfer them to someone else.
New bitcoins are created through a process called mining. But as the Bitcoin network has grown, mining has become too technologically complex to be practical for new users.
So for most people the better way to get Bitcoins is to purchase them from others who already have them. There are a number of ways to do this. There are Bitcoin exchanges, such as the European company Bitstamp, that match Bitcoin buyers and sellers. There are also user-friendly services such as Coinbase that will sell bitcoins to customers at the current exchange rate. The website Localbitcoins.com helps Bitcoin users find others in their local area interested in making face-to-face Bitcoin trades. Finally, in 2013 we saw the emergence of Bitcoin ATMs, which buy and sell bitcoins for cash. As of May 2015 there are around 400 Bitcoin ATMs around the world. Here’s a video of someone purchasing Bitcoins at an ATM: