All Bitcoin transactions are recorded in the block chain. A miner is a company that gets paid to collect new transactions into a "block" and add the block to the block chain.
The ability to add each block to the block chain is determined by luck. It is a kind of lottery. Each miner computes a number that is based on its block (called a "hash") over and over until some miner somewhere computes a winning hash. When a winning hash is found, everybody adds the winning block to their copy of the block chain and then all of the miners start competing for the next block. Typically, when a miner has computed a winning hash, we say that they have "found" or "solved" a block.
Of course, a miner that compute hashes more quickly is more likely to find blocks, but computing a hash uses electricity and computing more hashes costs more money, so only the efficient miners are able to make a profit.
A "mining pool" is a group of miners that work together and act as a single miner. If one of the members of the pool finds a block, then all of the members share the block reward.
The payment to the miner is called the "block reward". It consists of fees paid by the transactions plus the "subsidy", which is currently 6.25 BTC. To get paid, the miner simply includes a special transaction in the block that pays them the correct amount. This transaction is called the "coinbase" transaction.
New bitcoins are created by the subsidy. The initial subsidy was 50 BTC, but it is halved every four years approximately until it reaches 0 around the year 2140. Of course, as the subsidy is reduced, the transaction fees will become an increasingly larger portion of the block reward. When the subsidy reaches 0, no more bitcoins will be created and there will be about 21 million bitcoins.