Shigaraki Tomura's quirk, Decay, is a powerful emitter-type quirk that allows its user, Shigaraki, to break down anything he touches with all five fingers of his fingers. Decay has been shown to continue to break down whatever it begins to break down unless if it is forcibly deactivated, such as in the case of Eraserhead's elbow.
What is not known is how exactly Decay works. The working hypothesis that I have is that Decay works by breaking apart the molecular structure of solid matter. The reason Decay is restricted to solid matter is due to the fact that solid matter has a stiffer molecular bond that is easier to break apart rather than the looser molecular bonds between liquids or gases. This would explain why the decaying part of the body would need to be physically removed before the decay spreads to the rest of the body.
There is also the fact that Decay has no on or off switch. Rather, Decay is on all the time for Shigaraki, meaning he has to touch everything with four or less fingers or decay everything he touches (unless if it is actively being suppressed).
However, Shigaraki can not decay himself with Decay. In many points, we can see Shigaraki touching his hands together, meaning that he is not susceptible to his quirk. His clothes and severed hands are susceptible, though, as Shigaraki touches the hand over his face, Father, with care not to use all five of his fingers.
Therefore, Shigaraki Tomura's quirk, Decay, is an emitter-type quirk that activates whenever Shigaraki touches an object with five fingers and breaks the bonds between molecules that form solid objects.