Data availability answers the question, has this data been published? Specifically, a node will verify data availability when it receives a new block that is getting added to the chain. The node will attempt to download all the transaction data for the new block to verify availability. If the node can download all the transaction data, then it successfully verified data availability, proving that the block data was actually published to the network.
Data availability is critical to the security of any blockchain because it ensures that anyone can inspect the ledger of transactions and verify it. Data availability becomes particularly problematic when scaling blockchains. As the blocks get bigger, it becomes impractical for normal users to download all the data, and therefore users can no longer verify the chain.