Given a fingerprint (hash), you can't find any input that produces it.
The idea: A hash takes any input and returns a fixed-size fingerprint. Preimage resistance means the fingerprint can't be run backwards: given it, you can't find any input that produces it, except by guessing, which for a 256-bit hash means around 2^256 tries. That's what lets a system store the fingerprint of a secret instead of the secret, and lets hash-based signatures publish the fingerprints of one-time keys.
Why it matters: Protects stored passwords and some signatures.
Change one letter, change everything
Two inputs that differ in a single letter give completely different fingerprints: each square is one of the 256 output bits, red where the two outputs differ. About half the bits flip, with no pattern linking input to output, which is what makes running a hash backwards hopeless.
The details
Given a hash output, it is infeasible to find any input that produces it: a one-way function in practice[1].
It is the weakest hash property, needed for password hashing and for the one-time keys inside hash-based signatures. Grover's algorithm cuts 256-bit preimage security[2] to about 128 bits against quantum attackers.
Phillip Rogaway and Thomas Shrimpton. Cryptographic Hash-Function Basics: Definitions, Implications, and Separations for Preimage Resistance, Second-Preimage Resistance, and Collision Resistance. Fast Software Encryption, 2004. doi:10.1007/978-3-540-25937-4_24
Lov K. Grover. A fast quantum mechanical algorithm for database search. Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96, 1996. doi:10.1145/237814.237866