A layer 2 network is a blockchain system that runs on top of a base chain to handle transactions more quickly and cheaply, and the topic is covered in full under Layer 2. The word "network" is added when people mean a specific running system, with its own operators, users and tokens, rather than the general design approach.
Layer 2 networks process transactions away from the base chain, or layer 1 blockchain, and periodically post results or proofs back to it, so that they inherit much of its security. The main families are payment channel networks such as the Bitcoin Lightning Network, and rollups on Ethereum, which come in two forms: optimistic rollups, which assume transactions are valid unless challenged, and zero-knowledge rollups, which post cryptographic validity proofs.
In payments, layer 2 networks matter because many stablecoin transfers now take place on them, where fees are a fraction of base-chain costs. Users should note that moving funds between a layer 2 network and its base chain can involve bridges and waiting periods.