It's a game of yards... Progress is measured in yards. The attacking (offensive) team get four attempts (downs) to make ten yards. If they make it, they get another four downs to make another ten yards. This goes on from one side of the field, which is 100 yards long, to the other, 'till they get within ten yards of the goal line (end zone) at which point they get four attempts to score a goal (touchdown). Calls like "first and ten" mean your first attempt to get ten yards, "second and seven", "third and short" and so on... If it looks like they won't make the ten yards on their fourth attempt, they usually punt (kick) the ball into their opponents side of the field, turning over possession to the other team who now do exactly the same going the other way. Offensive teams are trying to make yards and defensive teams are trying to stop them and force a turn over. Good defences can drive an offence back so they end up having to cover a lot more than ten yards on a particular play. The "grab the ball" part you talk about is called the "snap". The attacking and defending teams line up face to face at the line of scrimmage (the point to which the attacking team carried the ball on their last play). The offence snap the ball to their quarterback (kinda like our flyhalf) who then passes or hands off the ball to whichever player is in the best position to gain yardage. The defence do everything they can to disrupt the quarterback and stop running or catching players.
Touchdowns score six points, a conversion is one, a field goal (like a penalty kick in rugby) is three points
Hope that helps Celine
EDIT: Should add, both teams have players who are better at defence and offence in their squad. When a team has the ball, their 11 best offensive players will be on the field. When they turn over the ball, the offensive players come off the field and the 11 best defensive players take their place.