In layman's terms, Bitcoin is a digital currency based on strong cryptographic principles, and designed to enable users to send money over the Internet without needing a credit card or bank account. Please see: http://en.wikipedia.org/wiki/Bitcoin
Who runs Bitcoin? What is the company behind Bitcoin?
In short, no one runs Bitcoin. Bitcoin is run collectively by the users who uses the Bitcoin Client, and any changes to the Bitcoin system have to be approved by the majority of users before they are implementated.
What is a Bitcoin address?
A Bitcoin address is a unique identifier which allows you to receive Bitcoins. With PayPal you send funds to an email address, and similarly with Bitcoin you send funds to a Bitcoin address. For example, this is an active Bitcoin addresses: 164RcnbMkNz53mfo3jAns28zHqi4v27LUG
You can also scan a QR code with your mobile device.
Where can I get free Bitcoins?
There are many places around the internet that will give you free bitcoins just for visiting websites. Each pay out a different amount of coins and distribute them on different days. Currently the best paying places to get Free Bitcoins are below.
Bitvisitor Most Popular
CoinAd Very Popular
BitCrate
Bunny Run
Many more available from this wiki page.
Where can I buy bitcoins? You can purchase bitcoinds online through several exchanges. You can also purchase bitcoins locally. Below are a few of the larger places of which to obtain bitcoins.
Mt.Gox
CampBX
Many more available from this wiki page.