Proof of Work
A system where computers compete to solve a hard guessing puzzle, and the winner adds the next block.
Miners repeatedly hash a block header with a changing nonce until the result falls below a target value, which takes enormous numbers of attempts but can be verified by anyone with a single hash computation. Because producing a block costs real electricity and hardware, rewriting past blocks means redoing all that work faster than the rest of the network extends the chain, so security is measured in the cost of the hardware and power an attacker would need. Difficulty adjusts so blocks keep arriving at the intended pace as mining capacity changes. Bitcoin popularized the design, and its energy consumption is a direct consequence of securing history through expenditure rather than a side effect that can be optimized away without changing the security model.
In practice
Bitcoin miners search for a header hash below the current target, and every full node checks the winning block with one hash computation.
The common misunderstanding
The puzzle does not compute anything useful; its only purpose is to be expensive to produce and cheap for everyone else to verify.