Inertia action, aka blowback

Delayed inertia, aka delayed blowback

Recoil action,  long and short

Gas

Combination

Blowback is limited by the cartridges it can handle as you need a stronger spring and or heavy bolt. This results in a heavy and/or hard to cock weapon. Ruger 10/22 is a blowback. MP5 is delayed blowback. They can be quite ammo sensitive, delayed action less so. Depending on the design that large heavy spring loaded bolt can increase muzzle jump and decrease full auto controllability.

Recoil action takes up a lot of space and the parts are generally beefy. Results in a large heavy but very reliable weapon.  Recoil operated weapons tend to have pronounced recoil over a gas operated design. Most semi automatic pistols of the Browning design are recoil operated. Recoil operated weapons require that they be firmly held so the action can reciprocate.

Gas operated guns generally use light weight reciprocating parts which allow high rates of fire. They tend to be more sensitive to ammo than recoil operated designs and require some way to regulate the amount of gas. Just about all modern military self loading rifles are gas operated.

Combination guns are usually special purpose pieces, the floating chamber used in the Colt Ace conversion is one example.

This leaves out purely mechanical driven designs like the chain gun and the rotary cannon.