Execution is the step in a resolution that ascertains whether a character succeeds or fails. In the IIEE model of resolution, it follows initiation (i.e. it is established that the character is trying to do something before execution is determined) and precedes effect (i.e. the impact on the game world can't be determined until the success or failure is known).
Almost all role-playing games have resolution mechanics that explicitly cover execution in some way, to the extent that it is often assumed that resolution mechanics are synonymous with the execution-step determination of success or failure. (Mechanics that focus on this are known as tests or checks.) It is rare for a game to leave out execution, either assuming that every attempt by a character is successful (or a failure) or by leaving the determination entirely in the hands of the GM.
Execution determines success or failure at a minimum, but may also determine the relevant degree of success, which will feed into the determination of effect.
Before the terminology of the IIEE model was settled, the execution step was sometimes known as completion.