An independent blockchain acting in concert with Bitcoin, Ethereum or other major chain, which retroactively became known as "Layer 1 chains" or "main chains."  Layer 2 chains process new transactions faster while reducing the load on Layer 1 and typically taking lower fees.
A connection (channel, bridge, etc.) is opened between Layer 1 and Layer 2, and at periodic intervals, a summary of Layer 2 transactions is added to Layer 1 for a permanent record.  A major consideration of Layer 2 chains is how transactions are validated before they are "cast in concrete" on the main chain.  New Layer 2 solutions are being devised all the time, making this is a major topic in the crypto world.  See 
blockchain.
Sidechains
Although sidechains and Layer 2 chains are often considered synonymous, their only similarity is that they function as auxiliary networks to a Layer 1 chain.  A sidechain has its own consensus method for adding blocks, and like a Layer 1 chain, a sidechain requires a sufficient number of validators.  See 
sidechain, 
consensus mechanism and 
Polygon blockchain.
Plasma Chains
Like sidechains, plasma chains have their own consensus method.  Plasma provides a framework for building Layer 2 chains on Ethereum.  At a fixed interval, a compressed representation of each block is committed to a smart contract on Ethereum.
State Channels (Payment Channels)
With state channels, crypto is deposited in a smart contract on Layer 1, and a connection is opened between two parties.  Payments are made on Layer 2, and when completed, a ticket is signed on Layer 1.  State channels can be bi-directional and also handle another party if channels have previously been opened.  Bitcoin's Layer 2 Lightning chain uses state channels (see 
Lightning Network).
Rollup Chains
Rollup chains accumulate and settle transactions.  At some interval, a summary of the transactions is "rolled up" and posted on the main Layer 1 chain.  Optimistic Rollups assume people are honest but transactions can be disputed (see 
Optimism).  Zero-Knowledge Rollups (ZK Rollups) provide cryptographic proof without divulging every detail (see 
zero-knowledge proof).  See 
Layer 3 blockchain.
Layer 2 Solutions
As Bitcoin and Ethereum became more popular, Layer 2 chains were developed to handle thousands of small-value transactions and store them as summaries on the main chain.  Layer 2 chains also keep Layer 1 chains from growing too large and cumbersome.