A hard fork refers to a radical change in the protocol of a blockchain network that effectively results in two branches, one following the previous protocol and one following the new version.
Lightning Network is a technological solution designed to solve the problem of transaction speed in the Bitcoin blockchain by introducing off-chain transactions.
Proof of Work (PoW) is a decentralized consensus mechanism that requires network participants to make an effort to solve an arbitrary mathematical puzzle so that no one can fool the system.