To add a bit to my earlier explanation...
Think about real money for a minute (real money for the purposes of this discussion is a conventional currency such as the pound sterling). Your bank balance is just a number in your bank's computers. The bank will turn it into notes and coins if you want but most of the time it is just a number. When you pay for something from your bank account, no actual physical money changes hands. All that happens is that the appropriate amount is deducted from your balance and added to the recipient's balance. It is entirely electronic. The same is true when you pay by credit card or debit card. There are, of course, safeguards in place to make sure that the amount received by the payee matches the amount deducted from your account.
There was a time when the pound sterling was backed by gold. You could, in theory, take your bank notes to the Bank of England and they would give you the appropriate amount in gold. That's what the promise on bank notes (I promise to pay the bearer on demand...) originally meant. However, we abandoned the gold standard a long time ago. If you take your bank notes to the Bank of England you will find that they will only change them for other bank notes (or possibly coins). Our notes and coins only have value because we all agree they do. Of course, they are also backed by a legal system that insists we must accept legal tender in payment of debts and by a central bank (the Bank of England) that makes sure the currency broadly retains its value.
In the same way, if you own bitcoin your balance is just a number on a computer. If you pay someone using bitcoin an amount is deducted from your balance and added to theirs. There are safeguards in place to ensure that the transactions match - your balance will go down by the same amount that their balance goes up. The only way your balance can go up without someone else's balance going down is if you successfully mine a bitcoin.
Just as with real money, bitcoin only has value because the people who use bitcoin agree that it does and are happy to sell goods and services in return for bitcoin. However, unlike real money there is no legal system propping up bitcoin nor is there a central bank tasked with maintaining its value. The exchange rate between bitcoin and real money is governed by what people are prepared to pay for bitcoin.
It may be that bitcoin enthusiasts are right that there is no need for a central bank. Personally, as I said previously, I am with those economists who think we are seeing a bubble and that the value of bitcoin will collapse at some point. But I may be wrong.
By the way, regarding traceability, DoctorTwo is partly right. The blockchain that backs up bitcoin records the date, time, IP address and wallet ID of every transaction. Each transfer is linked forward and backward to other transactions so it is possible to follow the trail. However, the blockchain does not record the user's identity or geographic location. If you don't want to be traced you can change your wallet ID before and after every transaction and it is also easy for anyone wanting to avoid detection to change their IP address. It is also possible to use something called a mixer to "wash" transactions. So it is certainly possible for people to make it very difficult for investigators to piece together their activities.
