Welcome to THR countryboy281
First off, this can be very confusing. There is no standardized system for naming a cartridge, that would be too easy. I tend to lump them all into 1 of 3 categories:
Metric: Probably the best system IMO, because it will often give you both the bullet diameter and the case size in the cartridge name. Example, 7.62x39 is a bullet diameter of 7.62mm (.30 caliber) in a case that is 39mm long. Metric will usually tell you if a case has a rim or not too, buy putting an "R" at the end of the name. An example would be 7.62x54R
Old Standard: These cartridges consist of 2 numbers, the first being the bullet diameter and the second being the power weight for the charge, in grains. .30-30 is the most common, a .30 caliber bullet with 30 grains of powder as the propellant. There are many others as well, .45-70, .30-40, .44-40, etc.
New Standard: Usually this consists of just the bullet diameter and the name of the company that created it (or standardized it from a wildcat). Example, .280 Remington is a .28 caliber (7mm) bullet that was standardized by Remington. While this is the most common name standard that you'll likely run into, I actually like it the least because it gives you very little info on the cartridge without having to do some research.
Of course there are some oddballs that don't always fit well into one of these 3 types, but they're the most common by far. And that's just for rifle/pistol. Shot shells are a whole different thing.
With shot shells, smaller = bigger. A smaller "gauge" is actually a larger shell. Example, the 20ga is actually smaller than the 12ga. The same thing is true for pellet sizes, a #7 pellet is actually smaller than a #4 pellet. So the general rule of thumb is that everything is backwards.
There are really 3 things to look for in a shot shell:
Gauge: This is the diameter of the shell. Most common are 12ga and 20ga, but there are several others.
Length: Some shells are longer than others. The standard is 2.75 inches, but some shells are as long as 3.5 inches. The barrel of your shotgun should tell you how long of a shell you can use.
Pellets: Again, smaller is bigger. Also pellets are broken down into 2 groups, birdshot and buckshot. Birdshot is much smaller and good for shooting clay targets or bird hunting. Buckshot is larger and is better for personal defense. Despite the name, I don't know anyone who hunts deer with buckshot. You can buy shot shells that have a single large projectile, they're called "slugs". If you hunt deer with a shotgun, you probably want to be using slugs.
Hope this helps... but probably just makes it even more confusing...