My favorite free rpg is The Window. This is a simple system in which characters assign dice to their skills. For example, my character might be “a crack shot with a rifle (D8)”. To accomplish something you roll your skill and try to get a 6 or under. If you are in a contest with someone, you both roll and the lower roll wins.
The Window is very simple are is made for narrative games. I enjoy it so much because it is easy to understand and I was once in a play-by-forum zombie game that used The Window as its ruleset. That game was a ton of fun. All of the players just focused on doing stuff; we never thought about stats or rules, We just role-played and had a ton of fun.
You can read and download the rules for The Window here.