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.