Proof of Work

1) Is a consensus mechanism that deters denial-of-service attacks and other service abuses such as spam on a network by requiring some work from the service requester, usually meaning processing time by a computer. (Wikipedia)
2) Is a system that requires a not-insignificant but feasible amount of effort in order to deter frivolous or malicious uses of computing power. (Investopedia)
3) Is a piece of data which is difficult (costly, time-consuming) to produce but easy for others to verify and which satisfies certain requirements. Producing a proof of work can be a random process with low probability so that a lot of trial and error is required on average before a valid proof of work is generated. Hash proofs of work are used for block generation. In order for a block to be accepted by network participants, miners must complete a proof of work which covers all of the data in the block. The difficulty of this work is adjusted so as to limit the rate at which new blocks can be generated by the network to one every 10 minutes. (Bitcoin Wiki)

Miners

1) Bitcoin mining is performed by high-powered computers that solve complex computational math problems (that is, so complex that they cannot be solved by hand, and indeed complicated enough to tax even incredibly powerful computers). When computers solves these complex math problems on the Bitcoin network, they produce new bitcoin. And by solving computational math problems, bitcoin miners make the Bitcoin payment network trustworthy and secure, by verifying its transaction information.(Investopedia)
2) Is the process of adding transaction records to Bitcoin’s public ledger of past transactions. The primary purpose of mining is to set the history of transactions in a way that is computationally impractical to modify by any entity. (Bitcoin Wiki)