The new team with the ball sends in its defense and brings out its offense. A touchdown occurs when the ball is caught by a player in the end zone, or run into the end zone by a player. Six points are scored. Yes, one extra point can be scored if the kicker can kick the ball through the goal posts. In football, the goal is to achieve yards toward the end zone.
Penalties punish teams by taking away yards that they earned.
Or, if the penalty is against the defense, the offensive team can be rewarded with additional yards. A Field Goal earns three points, and is scored when the kicker kicks the ball through the goal posts. This is not in conjunction with a touchdown. What happens is that, in many cases, the team is close to the opponent's end zone but is having trouble getting a first down. If a team is having trouble making a first down and they are too far from the goal posts to kick a field goal, they punt the ball toward the opponent's end zone.