Chains occur when panels clear and the resulting falling panels create other matches when they land. If those then cause other panels to clear, and it keeps going like that, the score really adds up, or the garbage block placed on your opponent becomes very large. You will be notified of each step in the chain by your character’s sound and a green box appearing with an ‘x’ and a number indicating how large the chain is (‘x2’ for the initial chain, ‘x3’ for the next step, etc.). Additionally, the sound of the next panels that clear will change for the first few steps, until ‘x5’ and later, which will sound the same as the ‘x4’ step.
Chains are fun to watch, but can take a lot of setup. For learning purposes, this is fine, but advanced players will tend to use Skill chains, because it’s much faster, and actually easier to visualize, when creating the chain as you go. There is no difference in the resulting score or garbage block however. This page will stick to the chains you can set up and then let go.