Here is a video that presents bitcoin mining, the most important activity connected with bitcoins.
Any person using bitcoins practically builds new currency. Miners are like the National Bank of bitcoins. Whenever a transaction is made, it is transformed into a complicated problem that has to be solved. The system was made especially hard, so the problems take a lot of time and resources to be solved. This way, the system makes sure that nobody can falsify bitcoins. Whenever a miner solves a problem, he or she receives bitcoins as reward. The transactions are verified, and other people are encouraged to mine also. As the transactions are more secure when they contain more code, more miners mean that transactions are safer.
The difficulty of solving the problems is changed by the network. More miners mean that for the moment, the problems would be solved easier for now, but also that the difficulty has to be increased, to encourage new miners to take the challenge.
In 2009, first miners used processors of their computers for that. Later, they discovered that graphic cards are more effective for this task. On the other hand, the effectiveness of those is given by the increased electricity consumption and heat. This is why the specialized ASIC computers appeared, which are made especially for mining bitcoins. Now, the mining process is faster and less consuming.
Many miners mean an increased difficulty and this is why standalone computers cannot solve those problems as fast as they used to. This is why pools were created, where different miners are able to solve the same problem as a team. Each member is rewarded considering his contribution.
Mining is the core of bitcoins, the system that makes sure that those transactions are secure, and that the network will grow.