A cryptographic failure is a failure of cryptography, or a failure of the implementation or use of cryptography. Essentially, it’s when something goes wrong with cryptography or something cryptography-adjacent.
In almost all cases, cryptographic failures are caused by issues in how cryptography is implemented or used, as opposed to a flaw in the cryptography itself. It’s not impossible for a cryptographic algorithm to have a hidden vulnerability, but it’s far less likely than improper implementation or use.
Cryptography can be used to maintain confidentiality, integrity, and availability; the three fundamental pillars of security. A cryptographic failure can break any one (or more) of those pillars, which is why cryptographic failures are so dangerous.