A Merkle proof shows that a specific value is included in a Merkle tree (a hash-based data structure) without requiring the verifier to see the whole tree. This enables efficient verification of large datasets - proof size grows logarithmically with dataset size.
Merkle proofs underpin many crypto primitives: airdrop eligibility (proving your address is on the snapshot list without uploading the whole list to-chain), light clients (verifying transactions without downloading the full chain), rollup state proofs, and more.