Bitcoin:
In 2009, a big problem was solved- the problem of how to spend cash on the internet.
Bitcoins, instead of being minted by a company or the government, are generated by powerful computers that spend all day verifying and compressing earlier Bitcoin transactions. When a miner figures out how to compress old records, it gets 25 new bitcoins. The computers are called Bitcoin Miners and anyone can buy or build one to connect to the bitcoin network, the most powerful computer network on earth.
To use bitcoins, you need a Wallet. There are dozens of options from Armory to Blockchain.info or Inputs.io. They're all free to use. You can even use your brain as a wallet by memorizing a passphrase that translates into a Keypair.
A Wallet is essentially two long numbers, an Address and Private Key. You can buy second-hand bitcoins on an online or local exchange like Satoshi Square in NYC or Bitstamp, Coinbase or Kraken online.
What this means for online commerce is that middleman companies like Paypal are getting closer to becoming obsolete. There are lots of other implications for the whole world, too.
What's a bitcoin worth? It depends on what someone is willing to pay for one. Coindesk.com has a ticker that tells you current exchange rates based on the online market. They're worth around $130 as of the time of this writing, but have been slightly past $200 in the past. Nobody knows how valuable they might be one day.
Bitcoin is:
P2P, which means that there's no middleman.
Open source, which means anyone can help build the program.
Cryptographically secure, which means that the transactions you make are encoded so nobody can intercept them.
Highly Divisible, instead of having two decimal places, like $1.00, bitcoin has eight, like this: BTC1.000000.
Optional fees, to speed up a transaction, you can pay a fee to a miner, but it usually isn't necessary.
There's lots to bitcoin. Visit bitcointalk.org or bitcoin.org to learn more.