The offset rule was introduced in TSU and has had many variants in subsequent games. In every implementation, this rule causes any chains made while nuisance is queued to fall on one's board to be 'offset' by the nuisance produced by the chain.
Rule Rensa Shibari (lit. Chain Requirement, a.k.a. Target Chain), which is typically available in multiplayer games also starting with TSU, makes it so you can only send trash after getting a certain number of chains. In games that have more complicated variants of the offset rule (i.e. SUN, YON and all Fever games), this fact is subtly important as it means the extra action triggered when offsetting does not occur even if you do make a chain.
Puyo Puyo SUN
In SUN, offsetting causes you to receive sun puyo which act as point puyo, except that they only increase your trash and not your score. An offset will cause the sun in the middle to bounce. After the sun stops bouncing, light beams appear on the field indicating the positions where the sun puyo will fall. The sun puyo will fall before spawning new puyo or after placing puyo on the field. On the turn when the sun puyo do fall, no ordinary trash will.
In YON, offsetting will give you additional charge for special attacks.
Puyo Puyo Fever
In Fever, offsetting will stop any queued trash from falling before the next drop comes, unlike in classic modes where garbage would fall immediately if all of it was not countered. Additionally, it adds a point to your fever meter. In Fever and Fever 2, it also adds a second to the opponent's fever time, whereas in 15th Anniversary, you gain your own time by making chains without offsetting.