In the most general incarnations, by pressing buttons in the order of weak, medium, and strong, normal moves have their recovery cancelled, performing sequential hits. As long as the direction is from weak to strong, the type of attack doesn’t usually matter; you may weave in punches and kicks, along with maybe even command moves, to include them in this chain.
This system was first implemented in the original Darkstalkers, where if you were to miss even a single part of the chain in the short timeframe it would be accepted, it wouldn’t happen. The actual name “chain combo” first appeared in Night Warriors and has been in the series since; in this incarnation, the damage correction was made stricter, but in return, input via hitting the same button rapidly and repeatedly (see mashing) became valid, and it was made possible to combo into a punch or kick of the same strength.
Also, in the Marvel series, some characters, regardless of the strength of the attack buttons, can chain from punch to kick or from kick to punch only once.
In Tatsunoko vs. Capcom, in which there is no kick/punch distinction, it is possible to chain from a standing attack to a crouching attack of the same strength (but not the other way around).
There are plenty of other games that have something similar to the chain combo system; these are often called “chain combos” as a generic term, even though they don’t always work in quite the same way.