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Cryptocurrencies employ the use of a technology called blockchain. In blockchain, time stamped information or records which are called blocks are linked to each other as they grow through the use of cryptography. This process then creates a chain of immutable records or information.
To add a block to an existing blockchain, miners need to solve a difficult mathematical puzzle with a high computer power or a computer that has a high processing power. But in a cryptocurrency using the Proof of Work concept, the next block is awarded to the first miner that solves the present mathematical puzzle.
Proof of Work concept is designed to prevent denial of services attacks and malicious blocks in the blockchain as you will need computing power more than 51% of the network(or 51% more than the rest of the miners computing power combined) to input a malicious block in the network or blockchain.
There is another concept which is called Proof of Stake which I will talk about later.
proof of work are the coins that require mining like bitcoin
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Not necessarily mining, you should watch out for my post on What is Proof of Stake.
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