Mining refers to the process PoW blockchains use to add new transactions to the blockchain. Miners compete against each other to validate batches of transactions by submitting the solution to a computationally rigorous mathematical puzzle, which is difficult to solve but easy for the rest of the network to check.
Staking seeks to deliver the same result: proving the validity of transactions. By first committing cryptocurrency as collateral, network validators are incentivized to behave honestly and penalized for not behaving honestly.