Proof-of-Work is a mechanism used by Bitcoin to achieve correct time-ordering of events within the blockchain, such that everyone can independently arrive at the same version of the Bitcoin transaction history, and therefore also the current state of bitcoin ownership, without the need for a central coordinator.
Most fundamentally, Proof-of-Work makes use of the laws of physics to prove the passage of time. Each new block in the blockchain must contain a proof that a certain amount of work was expended in its creation, and we know from physics that, if work has been expended, time must have passed. The work itself is expended by adding random inputs into a hashing algorithm, effectively guessing for a certain output. A correct output value, which could only have been found through repeated guessing, is therefore called a Proof-of-Work.
Each block is therefore proof that a certain amount of time has passed, and the sequential hash-chaining of blocks maintains an order to Bitcoin’s transaction record, proving that new blocks came after the preceding ones.
The difficulty of proving work, regulated by the Bitcoin software, is periodically adjusted so that the amount of work required to create a block reflects the ever-changing size of the mining network. This is targeted such that the total cooperative effort of all miners should generate a new block every 10 minutes on average.
Attaching a Proof-of-Work to each block enables anyone to objectively verify the relative ordering of each batch of bitcoin transactions throughout its entire history. Bitcoin’s fundamental consensus rule that the correct blockchain is the one with the most accumulated work, therefore allows all Bitcoin users to agree independently — in other words, without having to rely on any third party — which transaction record is the correct one.