A zero-knowledge rollup is covered in full under ZK-rollups (zero-knowledge rollups), which explains how they work, which networks use them and their trade-offs. This page explains the singular, spelled-out name, which often appears in research papers and project documents.
A rollup is a layer 2 network that executes transactions away from the main chain, such as Ethereum, then posts compressed transaction data and a summary of the new state back to it. In a zero-knowledge rollup, each batch comes with a zero-knowledge proof, often a zk-SNARK, showing that every transaction in the batch was processed correctly. A smart contract on the main chain checks the proof, which is far cheaper than re-running the transactions.
The main contrast is with the optimistic rollup, which assumes batches are valid unless someone submits a fraud proof during a challenge window. Because a ZK rollup's validity is proven up front, withdrawals to the main chain do not need that waiting period. The cost is complexity: generating proofs takes heavy computation, and building zero-knowledge versions of general-purpose virtual machines has been a major engineering effort. Networks such as ZKsync are examples.
"Zero-knowledge" here is something of a misnomer, since most rollups use these proofs for compression and correctness rather than privacy.