Suppuration is the process of pus forming. If you can't remove a tiny splinter from your finger, over time some suppuration may occur around it.
Use the noun suppuration when you need a nice clean medical term for the formation of pus, the white substance that sometimes oozes from a sore or a pimple. Suppuration can happen because of an infection or skin irritation. The Latin root is suppurare, "to form or discharge pus," from sub, "under," and pus, which literally means "matter from a sore," and figuratively, "bitterness or malice."