A one-way function is like scrambling an egg: easy to do, practically impossible to undo. Most of cryptography is built on the bet that such functions exist.
The idea: Almost every tool on the map has a one-way function as its engine. Store only the hash of a password, and a thief who steals it can't run it backwards. Feed a secret key and a counter into a one-way-style function and you get an endless stream that looks random: that's a cipher. Even signatures can be built from nothing else: publish the hashes of some secret values, and later sign by revealing some of the secrets. What one-way functions can't obviously do is let two strangers agree on a secret in public. That needs the extra structure of the public-key box.
Why it matters: Hashes, password storage, encryption and signatures all lean on this.
The details
On the map: alone they give PRGs, PRFs, MACs, symmetric encryption, signatures, commitments, ZK for NP (Minicrypt). Key exchange and public-key encryption need more: see the public-key box. Equivalent: time-bounded Kolmogorov complexity is mildly hard on average (Liu-Pass 2020)
A one-way function is easy to compute but hard to reverse: given the output, you can't find any input that produces it. Think of mixing paint: easy to mix, practically impossible to unmix.
It is the minimal assumption behind computational cryptography. From any one-way function you can build pseudorandom generators[1], pseudorandom functions[2], symmetric encryption, message authentication codes, digital signatures[3] and commitments[4]. Impagliazzo called this world "Minicrypt"[5].
What one-way functions alone can't give, at least in any black-box way[6], is key exchange or public-key encryption: that needs more structure (see the key-exchange node). Nobody has proven one-way functions exist (that would prove P ≠ NP). Liu and Pass (2020) showed they exist exactly when a 1960s problem, computing time-bounded Kolmogorov complexity, is mildly hard on average[7].
Johan HÅstad, Russell Impagliazzo, Leonid A. Levin and Michael Luby. A Pseudorandom Generator from any One-way Function. SIAM Journal on Computing, 1999. doi:10.1137/s0097539793244708
O. Goldreich, S. Goldwasser and S. Micali. How To Construct Random Functions. 25th Annual Symposium onFoundations of Computer Science, 1984.. doi:10.1109/sfcs.1984.715949
J. Rompel. One-way functions are necessary and sufficient for secure signatures. Proceedings of the twenty-second annual ACM symposium on Theory of computing - STOC '90, 1990. doi:10.1145/100216.100269
Moni Naor. Bit commitment using pseudorandomness. Journal of Cryptology, 1991. doi:10.1007/bf00196774
R. Impagliazzo. A personal view of average-case complexity. Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference, 1995. doi:10.1109/sct.1995.514853
R. Impagliazzo and S. Rudich. Limits on the provable consequences of one-way permutations. Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89, 1989. doi:10.1145/73007.73012
Yanyi Liu and Rafael Pass. On One-way Functions and Kolmogorov Complexity. 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS). doi:10.1109/focs46700.2020.00118