Bitcoin mining is the process by which new Bitcoins are generated. When you perfom mining, your computer adds new Bitcoin transactions to the block chain (a public ledger where all Bitcoin transactions are stored) and searches for new blocks. A block is a file that has the most recent Bitcoin transactions recorded in it. When your computer discovers a new block, you receive a certain number of Bitcoins. Currently a block contains BTC 25. This number changes throughout time and gets smaller by the factor 0.5 every four years.
Since bitcoin mining has become a hardware intense and therefore expensive process, most individual miners join a so called mining pool. One of the mining pools you can conect to is BitMinter for example. By providing computing power to their pool you can earn Bitcoins from mining without the need to build your own big mining farm. There are entire communities around Bitcoin mining and besides the fact that you earn Bitcoins it's also fun. You meet new people online and get in-depth knowledge about Bitcoin as a protocol and technology.
If you want to earn Bitcoins through mining, be aware that it is a costly and time consuming process. Read the respective introductions and manuals to learn more about it. This website is a good starting point. Unless you are mining just out of curiosity and want to get to know the technology, it is important to make a cost / benefit analysis. Hardware prices, electricity costs, bitcoin difficulty and the Bitcoin value influence the profitability of Bitcoin mining. If all this seems interesting to you and you want to earn Bitcoins from mining make your first calculations on the Mining Dashboard.
