1. Zero-knowledge
That's proving a statement without revealing information, this is the example your're giving. For instance, you could prove that you have an account number (private information, input to proof generation, but not input to proof verification, so not revealed to the counterparty) on a blockchain (whose state hash is public, input to both proof generation and verification), with enough quality assets to back your loan.
2. Succintness
It turns out you can verify zk-SNARKS (where SNARK = Succint Non-interactive ARgument of Knowledge) in less time than it takes to execute the program being proven. Some schemes do this in constant time, some in logarithmtic time (the STARKs currently used with Cairo are logarithmic).
This lets you offload computation to a third party (like AWS) but without trusting that the third party will not perform a modified computation instead. The third party gives you back the result along with a cryptographic proof that you can verify relatively fast.
In this use case, there doesn't need to be any private input at all (though there can be, and the verification will still be succint), so it's not even necessarily "zero-knowledge". People sometimes call this "validity proof" to distinguish, but the underlying cryptography is the same.