I saw a person put thier logo on a game that they hadn't made, how do you do that, I have permission from the game developer to do it but he didn't show me how.

What do you mean "put your logo on the game"?

Anyway, the short answer to your question is: it depends on the game. All games are different and generally have their own engine and code. So there cannot be one program to edit all games. Games that are editable ship with an editor that allows you to modify many things about the game: create new levels, put in your own graphics, change the AI, change the tuning, etc. These are known as mod-able games.

Another route is hacking/cracking the game; sometimes it's possible to do things with a game, sometimes it's now. However, since you don't already know how to do that, it's at least 5 years of hard work and skill learning away from you (you need to learn to program first which will take a while). So no point in describing the generalities of how it's done.

[EDIT: also, if you have permission from the game developer, then ask them how]

-me