A field goal can happen on any down and from anywhere on the football field. A place kicker will line up behind the scrimmage line and kick the ball through the goal post. A team should be in field goal range before they go for a field goal. Depending on the scenario, some teams can choose to make a field goal attempt from more than 30 yards back.
The extra point or PAT happens after a touchdown is scored. The kicker must kick the ball through the goal post for one additional point. This kick is performed by a special teams kicker, who's main role is to make field goals. The extra point can be a big determinant for who wins the game.
A punt is used to change possession. Punts are usually made on 4th down to prevent a turnover on downs. A high punt allows the defensive players to gain yards down the field and set up defense so the catcher can't make any progress.
A place kicker is a type of kicker that performs kickoffs, field goals, and extra points. The ball is placed on a tee or held in place by a holder, a player on the special teams. Placekickers is the most common term for any kicker on the team that performs a special kick.
A punter is a type of kicker that performs punts. Punts change possession to the other team and are an important change of pace in the game. The punter must be a very strong kicker and have very good flexibility. This kick should be powerful enough to clear the field so the other team is as far down field as possible.
A drop kick is a type of kick that happens on punts. The kicker gets the ball from the snap, drops it in the air and kicks it. The drop kick requires great hand eye coordination and a lot of power.