A bullet isn't centerfire. A cartridge can be centerfire or rimfire. The bullet is the projectile that goes down the barrel. A cartridge is the combination of bullet, casing, propellent powder and a priming device. A centerfire cartridge has a primer in the center of the case head.
Boxer style centerfire primer
Berdan style primer
A centerfire cartridge e.g. 9mm or .308
A rimfire cartridge e.g. .22LR. The primer is in the rim of the cartridge.
Most handguns and hunting rifles use a centerfire cartridge. The illustrated centerfire primer is the "Boxer" style. It can be reloaded by poking out the expended primer and replacing it. The other style of primer is "Berdan". The Berdan primer has an "anvil" in the center of the primer and "flash holes" around the edge. Poking out an expended Berdan primer is moderately difficult. In general, you won't find Berdan primers for sale for the purpose of reloading an expended cartridge.