Yes, there are a lot of bullet sizes and it can be confusing. We call them “calibers,” except for shotguns, which are referred to as “gauge” or “bore.” Each is different and — just as with anything else — each has certain attributes that are beneficial for certain purposes…and its own drawbacks when the same caliber of cartridge is used for a different purpose.
How a round’s caliber is determined is usually from the diameter of the bullet (meaning the projectile at the end.