Centerfire ammunition is any cartridge which has the primer seated in the center of the casing head, as opposed to rimfire ammunition. Most modern ammunition are centerfire cartridges.
Centerfire ammunition can either be rimmed or rimless; rimmed ammunition (where the case rim diameter exceeds that of the case itself) is mostly found with older bolt-action and lever-action rifles, shotguns, and cartridges intended for use in revolvers. The rim aids in positive extraction, and keeps the round from going too far into the cylinder on revolvers, which would bind the action.
Rimless ammunition isn't actually rimless; the rim diameter on rimless cartridges does not exceed that of the case itself. This type of ammunition is found mostly with modern automatic weapons, semi-automatic weapons, and bolt-action rifles, as they are more easily fed through box magazines. Weapons used with this type of ammunition have extractors designed to catch the smaller rim.