The two completion rule awards a first down to the team that completes two forward passes. In order to get a new set of downs, you have to complete two passes. Running plays are useless. Yes, you can run, but running does not contribute to the two completions, hence teams are less likely to use them.
The two completion rule usually provides the impetus to throwing a lot of passes and a high flying offense. Because of this rule, teams pass, pass, and pass some more. The passing game really takes off. Running is only used sparingly to surprise the other team. Usually there are no linemen required and all the players on the field are skilled players; either a quarterback, wide receiver, and occasionally a running back.
Using the two completion rule usually works when you are younger and developing your football skills. Normally this rule is used up until players graduate to more advanced levels of play. While the two completion rule allows you to improve your passing game, it severely limits the offense and is not realistic. Players eventually realize this and graduate to a more advance way of playing backyard football.
Eventually, defenses that play man to man coverage will figure out how to stop this pass-only offense and the games will not be as exciting. As you graduate to organized forms of football, peewee and middle school football, this two completion rule becomes less and less appealing and less realistic. You will quickly realize how uninteresting this style of play is.
A pass-only offense is easy to limit and defend. Eventually, adding running plays will open up the passing game, but that’s another article for another day.