How does it work?
If you put the transaction into proof of work decentralized consensus-based system, you need to verify the transaction. The transaction can be verified only by a miner, which must solve computer algorithm (needs computing power) called client puzzle protocol (CCP). Only thru miner verification transaction can be processed and sent to another user of the network. Also, you can look at Wikipedia for the deep explanation.
Miners and mining
Miner is the person, who operates mining equipment. Miner quest is solving CPPs and verifying transactions in the network. As a result of his services, he is rewarding with transaction fees, which also means the gradual release of the currency into circulation (For example BTC has limited supply, which is gradually released into the network thanks to the mining).
It is also necessary to mention, that proof of work consumes a huge amount of electricity power. For detail you can check Bitcoin Energy Consumption Index. As you can see Bitcoin eletricity consuming is huge problem and it is approaching to some states. The higher the price of the Bitcoin will be, this will require more energy for network running.
Usage of Proof of Work
Proof of Work is probably the most proven solution in crypto space (by the time). You can find it in Bitcoin, which is life since 2009. Also, you can find it in Litecoin, Monero and a lot of others. Longterm view for proof of work is very demanding for operation world payment network.