Before we dive into what state channels are, let's take a look at the overall ecosystem and why they are necessary in the first place. At a high level, blockchains suffer from "scalability" issues. By modern standards, blockchains are very slow and comparatively expensive to run. For comparison, Visa is capable of processing around 50,000 transactions per second while Ethereum can only handle a maximum around 20.
This gap creates an urgent need for technologies to help scale blockchains. It is essential, however, that these technologies preserve the security and decentralization of the blockchain--otherwise, applications might as well use legacy technologies.
Projects are tackling the scalability problem in lots of different ways. Some of these projects are attempting to make blockchains more scalable by upgrading the blockchains themselves. This is known as “layer 1” scaling because it modifies the base blockchain. Other projects are trying to build on top of existing systems. We call this “layer 2” scaling because we’re adding a new layer and not changing the underlying system.
Most layer 2 solutions boil down to a simple observation: not every interaction needs the security and decentralization of the blockchain. If you're transacting with a store, for example, what matters is that you and the store agree on what you were charged. Later on, you and the store can transfer funds on the underlying blockchain, but both of you will already know exactly how your balances will need to change.
State channels are a type of Layer 2 solution focused specifically on this sort of 2-party interaction. Put simply, they let users transact with each other off the blockchain, so long as they come to a consensus about the transactions that are occurring. If they can't agree, they can use the blockchain as the ultimate arbiter of who owes how much money. In this way, state channels maintain the security and decentralization of the underlying blockchain.
As a result, these transactions can happen very rapidly and very cheaply. If you'll recall, the two biggest blockchain scalability issues are cost and transaction speed; for interactions between two parties, state channels can solve those issues.
A great resource for understanding Layer 2 solutions and the differences between them is Josh Stark's Making Sense of Ethereum’s Layer 2 Scaling Solutions.