A layer 2 solution is designed to help scale applications by handling transactions off the blockchain mainnet (layer 1), while taking advantage of its high security.
In essence, a layer 2 solution is a protocol built on top of an existing blockchain in order to attempt to address problems of transaction speed and scalability of the main blockchain.
For example, the two main cryptocurrencies, Bitcoin and Ethereum are still not able to process thousands of transactions per second, even as their user base increases. In this instance, a layer 2 solution would refer to the multiple solutions being proposed to solve this problem. Two major examples of layer 2 solutions are the Bitcoin Lightning Network and the Ethereum Plasma.
Layer 2 solutions are a secondary network, where transactions can take place independently of the mainnet, which means they can also be referred to as off-chain solutions.