Each weapon has its own unique execution. Earning the prerequisite amount of kills will make that weapon's execution available for use in Versus.
Seeing as how there were little to no individual weapon execution videos, here's a video from YouTube user calloftreyarch performing every possible execution in Gears of War 3 (including those needed for the Judge, Jury, and Executioner achievement).
Starting Weapon Executions Unlock After 200 Kills:
Pick Up/Power Weapon Executions Unlock After 40 Kills: