Proof of Work is system or protocol, which prevents a network (Blockchain) from possible cyber attacks like DDOS (distributed denial of service), Spam or fraud actions. Simply said, Proof of work protocol provide verification, that all transactions in the network are real through mining. Proof of Work was first concretely described in 1999 but he gained popularity after the use in Bitcoin with a connection to the Blockchain technology.
Generally, computers, which are working in the same network cannot trust to each other, because one of the computers can run some fraud actions for profit or network damage. It is necessary to define the system, how can computers communicate between themselves and trust to each other without any possible fraud. This solves proof of work system based on blockchain and decentralized consensus.
Decentralized consensus means that one member of the network can not control all blockchain network. Also, it is easy to explain, that if you want to send money to your friend you do not need to trust any third party like Visa, PayPal. If you are sending money standard way, you have to trust that Visa, PayPal (3rd party) will transfer money to your friend without any abuse your personal data.
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.
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.
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.
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