Layer 2
A separate network that processes transactions off a base chain while relying on that chain for security or dispute resolution.
The general idea is to move execution somewhere cheaper while keeping the base chain as the place where assets are held and disagreements are settled. Rollups do this by posting their transaction data and a state commitment back to the base chain, so anyone can reconstruct the layer 2 state and challenge an incorrect result; payment and state channels do it by locking funds in a base-chain contract and settling only the opening and closing balances, as in Bitcoin's Lightning Network. Fees are far lower because the base-chain cost of a batch is shared across every transaction in it. How much security is actually inherited varies enormously, and the label is applied loosely, including to systems whose base chain cannot verify or challenge anything they do.
In practice
Bitcoin's Lightning Network settles a channel on the base chain only when it is opened or closed, while payments inside the channel never touch the chain at all.
The common misunderstanding
Not everything labeled layer 2 inherits the base chain's security; if the base chain cannot verify or challenge the layer's state, users are trusting the operators instead.