This is the process of storing transactions in the blocks of the blockchain, while verifying their validity. When a block is filled, certain amount of new coins is created and awarded to the miner who finalized the block.
The time it takes to complete a block and the amount of coins awarded to the miner is fixed, no matter how many miners are involved in the process, just the difficulty of the process goes up ... so the more miners, the higher computing power is actually used to complete a block. As a result, it costs more and more $$$ in order to complete a block, hence increasing the "production" cost of a single coin.
Currencies that depend on this mining process are called POW (Proof Of Work) Coins. Bitcoin was the first currency to utilize this approach.
In order to mine effectively, you need to use specifically designed mining devices. The most popular are Mining Rigs build by using high end graphics cards or specific mining devices called ASIC Miners.