The process of cryptographically computing a mathematical proof for a block, containing a number of transactions, which is then added to the blockchain.
Mining is the distributed process of confirming transactions on the public ledger of all transactions, aka blockchain. Monero nodes use the blockchain to distinguish legitimate transactions from attempts to re-spend coins that have already been spent elsewhere.
Monero is powered strictly by Proof of Work. It employs a mining algorithm that has the potential to be efficiently tasked to billions of existing devices (any modern x86 CPU and many GPUs). Monero uses the CryptoNight Proof of Work (PoW) algorithm, which is designed for use in ordinary CPUs and GPUs.
The smart mining feature allows transparent CPU mining on the user's computer, far from the de facto centralization of mining farms and pool mining, pursuing Satoshi Nakamoto's original vision of a true P2P currency.