What is proof-of-work?
The idea of proof of work (PoW) was first explained in a paper Cynthia Dwork and Moni Naor [1], though the term "proof of work" came later [2]. It was first proposed as a way to deter spam, but it's better known these days through its association with cryptocurrency.
If it cost more to send email, even a fraction of a cent per message, that could be enough to deter spammers. So suppose you want to charge anyone $0.005 to send you an email message. You don't actually want to collect their money, you just want proof that they'd be willing to spend something to email you. You're not even trying to block robots, you just want to block cheap robots.
So instead of asking for a micropayment, you could ask the sender to solve a puzzle, something that would require around $0.005 worth of computing resources. If you're still getting too much spam, you could increase your rate and by giving them a harder puzzle.
Dwork and Naor list several possible puzzles. The key is to find a puzzle that takes a fair amount of effort to solve but the solution is easy to verify.
Bitcoin uses hash problems for proof-of-work puzzles. Cryptographic hash functions are difficult to predict, and so you can't do much better than brute force search if you want to come up with input whose hashed value has a specified pattern.
The goal is to add a fixed amount of additional text to a message such that when the hash function is applied, the resulting value is in some narrow range, such as requiring the first n bits to be zeros. The number n could be adjusted over time as needed to calibrate the problem difficulty. Verifying the solution requires computing only one hash, but finding the solution requires computing 2n hashes on average.
Related posts[1] Cynthia Dwork and Noni Naor (1993). "Pricing via Processing, Or, Combatting Junk Mail, Advances in Cryptology". CRYPTO'92: Lecture Notes in Computer Science No. 740. Springer: 139-147.
[2] Markus Jakobsson and Ari Juels (1999). "Proofs of Work and Bread Pudding Protocols". Communications and Multimedia Security. Kluwer Academic Publishers: 258-272.