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.
As Bitcoin is a peer-to-peer network, shutdown in this context would need to mean the closure or abandonment of the P2P network. If this occurred and insufficient peers attempted to recreate the network - then yes, you could consider the network "shutdown".
In which case your Bitcoin money has no value and essentially disappears in a puff of smoke. This would be similar to if paper notes suddenly super-glued themselves to people's hands, pockets or purses.
This scenario isn't entirely esoteric.
Consider the fact that the energy required to process on-blockchain Bitcoin transactions can only ever increase (see assumption below). If the peer network massively contracted, the velocity of the currency would fall through the floor due to insufficient network processing power. And unlike the initial speculative rush that bootstrapped the crypto-currency when "energy per hash" was cheap, the only people with the incentive to re-establish the network would be those with significant trapped Bitcoin reserves and enough non-Bitcoin money to pay transaction fees for miners to establish ASIC farms.
Paying these transaction fees to re-inflate the network would be an uphill battle; as these fees would start very high (i.e. energy cost per peer) and only decrease if enough people paid those high fees to amortise the mining costs and draw the rest of the populace back into the network to keep the fees cheap and stabilise the ROI for block mining.
Escalating difficulty is the core assumption of this scenario: That energy per hash can't go backwards due to a risk of "unwinding" existing transactions.
Bootstrapping the new network with a lower target might work - but who owning mature Bitcoins wants to re-enter the initial speculative rush due to a low target? You would need to rework the network protocol to not create new Bitcoins on new blocks until the target re-reaches the previous network's difficulty level.
Whoever controls the private keys, controls the money
Bitcoin relies on two keys: a public one, and a private one. The public one is used for receiving funds, the private one is used for spending them. There is one public key for every private key. You are encouraged to create fresh keys for every transaction to preserve your anonymity.
Securing your private key is right at the top of your todo list because if someone copies that, then they can spend your bitcoin. The best way to secure your private key is to keep it on trusted hardware, the next best is to use a HD wallet. More about these later.
Bitcoin is peer-to-peer. This means that there is no central site to shut down. If you are in complete control of your private keys then there is nothing to threaten your money. Your funds are safe.
If you trust someone else to look after your private keys for you - such as some online wallets - then you're trusting that they won't spend your money, or get shut down.
To protect themselves from various attacks, most online wallets ensure that the private keys remain encrypted (by you) so they can't spend the money. However, if you don't have backups of those private keys and the online wallet site is shut down then you lose access to the private keys and therefore your money.
So what were these HD wallets then? These are the next generation of Bitcoin private key management. They are essentially brain wallets on steroids and are much more secure than the classic encrypted wallet approach.
Many of the other answers provide some useful information, but the spirit of the question seems to be: "If I have to use [website name] to access my wallet, which I have chosen from a list of competing service options, what happens if that website goes offline?"
What's important to remember is that the wallet is not actually on any of these websites - or even on your computer if you create one locally. Another answer suggested that the solution is to never use a website and only use a computer, but that doesn't properly address the question because it could just be rephrased as "what happens if my computer burns up in a fire along with all my backups"
The answer is that many websites, along with offline programs, allow you to import/export you wallet. These are all the same address so they synchronize together. If it is an encrypted wallet you need to know the pass phrase that was randomly generated when the wallet was created. In the event that you don't know the pass phrase the wallet is gone forever. If you have the pass phrase, but never bothered to create a backup, there are steps you can follow that will allow you to recover the information needed to import to any other wallet service and continue.
So to answer the spirit of your question: As long as to take some personal responsibility to secure your own information the wallet will remain accessible to you even if a given service goes under.
There is an answer over here that goes into detail on how to recover your wallet.
In the event that you neglect to secure your wallet, and the service goes under, the Bitcoins still exist, but they will never be used by anyone ever.
If you are still concerned you could use the paper wallet feature. As explained here, this involves taking your Bitcoins offline and storing on a code on paper just like dollars. Of course, this makes them as secure (or not) as real physical dollar bills.
Assuming you are using your own wallet: Nothing happens. Bitcoin is peer-to-peer. It does not rely on a central server or website. You still have your bitcoins and your wallet is still working. The bitcoin peer-to-peer network is still operating and you can still transfer coins.
If you are not using a bitcoin wallet that you control yourself, but storing the bitcoins at a website (in someone elses wallet), then if they shut down the website and run away with your bitcoins, that would mean the coins are lost. This has happened with several scams in the past.
If you mean the bitcoin.org website and other sites with wallet software:
In the extremely unlikely event that all websites hosting bitcoin wallet software could be shut down, then it could make it more difficult for new users to get the necessary software. This might last a few minutes and be a minor inconvenience for new users considering bitcoin.
Apparently your father is mixing up Bitcoin with Facebook. Bitcoin is not a website. For this purpose Bitcoin is not even web based but another internet service (the terms web and internet are not synonymous).
The network, as a distributed entity carried by all participants, is usable for you as long as you can retain a connection to a single remaining peer, though ideally you would prefer to have connections to multiple peers. Your ability to connect to other peers is not controlled by any central authority as it is the case with, i.e. Skype. "The Bitcoin people" for that matter are all participant in the network, including yourself.
Other than with classical web based services there is no single point of failure.
If your connection to the last peer fails or if the last peer goes down beacuse the network went out of fashion, you and others will still keep the contents of your Bitcoin wallets and a copy of the global transaction chain as it was at the latest time of your participation. Once any nodes come up again in the network they can continue running from this point.
This is why you hear and are told:
"web wallets are bad" Lesson, keep your own private keys! (if possible, generate many many blank paper wallets with a computer not connected to the internet. Send spendable bitcoin to a 'hot wallet' and send your change over to a new paper wallet and throw away the original [you have no way to 'prove' that the private key scanned or typed in was not compromised upon use])
His question is answered with another question - What happens if paypal servers got shut down, does the money disappear?
Most likely the DB or wallet software implementation they used stored all the private keys for the balances on the wallet server. If it was on a type of virtual instance and that instance had no backups and it were to be deleted, then the balances would remain there, on the blockchain, but without those private keys, those balances would remain unspendable and effectively useless.
Your TIL moment should be: keep bitcoin across different balances and mediums (its possible to keep them stored on physical pieces of paper and metal too). Don't trust an online 3rd party like circle or coinbase with large sums of your money.