Q. What is Bitcoin
A. Bitcoin is a digital currency. It is a currency, like Dollars, Euros, Pounds etc but it only exists in digital form - you cannot get actual coins and paper notes. It is used by thousands of people all over the world every day to buy and sell stuff. There is no central authority that controls Bitcoin, it is a decentralized, global currency that relies on cryptography to ensure the integrity of your transactions.
Just like other currencies, it can be exchanged on a currency exchange. The biggest one that handles vast majority of Bitcoin trade is MtGox.com You can see the current exchange rates there.
There are many places where you can learn more about Bitcoin, such as here, here, here and here.
Q. How can I start using Bitcoin?
A. First you need to download and install a wallet. Here are several popular choices:
Then you need to buy Bitcoins and transfer them to your wallet. Here are some places where you can buy Bitcoins:
BitInstant.com - Buy Bitcoin for cash at thousands of locations worldvide.
CoinBase.com - Buy Bicoin through your bank account
LocalBitcoins.com - Buy from local people in your area
MtGox.com - Buy and Sell Bitcoin at the online currency exchange
Here is a nice guide to getting started which also has many more links to places where you can buy Bitcoins:
Q. Why should I use Bitcoin?
A. While Bitcoin is not 100% anonymous (although, with a bit of care on your part, it can be) as a determined person could in theory and with a lot of effort, find your IP address, it affords a lot more privacy than other online payment methods. You do not have to provide your name, address or any other personal details to make a purchase with Bitcoin.
It is available in any country. If you live in a country from which it is dificult to make online purchases (for example due to credit card companies policies) Bitcoin is for you.
There is no risk of identity theft.
No fees for transactions (except voluntary).
You have full control of your money from anywhere in the world 24 hours a day, 7 days a week.