A quarterback in flag football is the same thing as a quarterback in any other form of football: they control the football on almost every play and they often run the offense. The quarterback can run and pass the football and they often call plays for the offense. The quarterback is often the most important player on the team; this can also be said for most other forms of football.
A quarterback can run the ball in football. Any player on the field can run the football, the quarterback is no exception. The quarterback only usually runs if he is in danger of being sacked or tackled behind the line of scrimmage, which would be a loss of a down and yardage. Sometimes teams with mobile quarterbacks will run 'designed runs' in which the quarterback may run the football on purpose. This strategy isn't implemented very often as it risks the health of the quarterback, who is often the most important player on the field at any given moment.
There are also some rules in place to protect quarterbacks, given how important they are to a team and sometimes difficult to find and/or replace. There are certain things that defensive players cannot do to the quarterback during the play. Similar to regular tackle football, the defensive player is not allowed to run into the quarterback behind the line of scrimmage. If a defensive player makes contact with the quarterback's arm or hand, it is a penalty. Even if the defensive player made contact with the football first, which typically gets a player out of a penalty in tackle football.