What happens to bitcoins if the site shuts down? My father just asked me:  What happens if the bitcoin people decide to shut down the website, what happens to my bitcoins?  I didn't know.
The ledger of Bitcoin balances is stored on a network comprised of thousands of individual computers around the world. Anyone can run the Bitcoin program which contributes to that network and stores the record of every transaction that has ever happened. This is what is called a peer-to-peer network or a decentralized network.

There is no website that controls Bitcoin, but there are websites that pull information from the network and present it to you. You can trust them to store and report to you that data, but using the program itself on your computer is the most secure way to handle your funds, and it has the added benefit of also securing the network a bit more.

There are many websites which do the work of downloading the program and managing your Bitcoins for you - in the same way that paypal manages your money on their website. If you trust paypal to not decide to "freeze" your funds for no apparent reason, which they have MANY TIMES before, you would deposit money into their website. It's exactly the same with a Bitcoin website that stores your funds. However, a very notable benefit of Bitcoin is the ability to securely store your coins on your PC, which is not possible with cash. Additionally, as long as you hold the wallet file or technically the private key that is assigned to the program on your PC, your Bitcoin can never be frozen by anyone. The only requirement for spending those Bitcoin in the future is that you have your wallet file (or your private key) and that the network still exists. Currently the network is only getting bigger as the days pass.

There are questions that follow after understanding this answer, such as how transactions are validated if there is no central authority, and there are considerably sufficient answers to those questions elsewhere on this forum and other places online.

I suggest you check out https://www.weusecoins.com/en/questions -- it has the answer to many of the preliminary questions you may have, and the main page also has a quick video which does a good job of explaining the concepts to newcomers.