[ahomescu1]

Every time a block is added to the block chain, whoever "solved" that block gets a fixed number of new bitcoins (25 is the current number, I think).

A block contains several pieces of information: a list of transactions, a link to the previous block, some other stuff and a nonce. The nonce is a value used by the mining process. "Solving" a block means finding the value for the nonce that, together with the proper contents of the block, hashes to a hash value that starts with a given number of zeroes. You can't really find a correct nonce any other way than by trying new nonces until you find a correct one, hashing the block each time. This is what the mining process is, in short.