With government-issued currencies (fiat currencies) like the USD or EUR, a central authority such as a government or a bank has the responsibility of keeping track of all the currency in circulation, making sure that third-parties cannot issue new currency.
With Bitcoin and other cryptocurrencies, miners replace the role of the central bank, securing the network and making sure no new coins are issued or that existing ones are double-spent. Miners are participants on the blockchain who contribute resources such as processing power (CPU) to help maintain the list of blocks and the transactions within them. They do this by solving complex mathematical puzzles, validating the authenticity of the transactions (as someone had to spend real world resources to do so). As each new block is mined, a small, new amount of the currency is generated which serves as the reward for miners.
The mechanism described above is formally known as Proof of Work (PoW), and commonly known as Mining. While PoW was one of the earliest implemented types of mining, there are a number of others present in the public domain today, such as Proof of Stake (PoS) and Proof of Elapsed Time (PoET).