Mining is the process of creating a block of transactions to be added to the Ethereum blockchain. Miners essentially process pending transactions and are awarded block rewards in the form of Ether, the Ethereum network's native currency, for each block generated. Generating a block requires intensive computational work (or hashing power) due to the difficulty set by the Ethereum network protocol. This difficulty level is proportional to the total amount of computational power (also known as the total hashrate of the network) being used to mine Ethereum and serves as a way to secure the network from attacks as well as tuning the speed at which blocks (and block rewards) are generated. This system of using hashing power generated by costly computer hardware is known as Proof of Work (PoW).
Technically, anyone is able to mine on the Ethereum network using their computer. However, not everyone is able to mine Ether profitably. In most cases, miners must purchase dedicated computer hardware in order to mine profitably. While it is true anyone can run the mining software on their computer, it is unlikely that the average computer would be able to earn enough block rewards to cover the associated costs of mining (See question below for more details).