Overtime games are very common in the NFL, and teams practice the plays they would use in an extra time situation on a regular basis.
Whereas the NFL overtime rules used to dictate that overtime periods be played in accordance with a "sudden death" format (a score of any kind automatically wins the game), the rules were adapted in 2017 to make overtime more fair and exciting.
A coin toss decides which team will receive the ball to begin an overtime period. The offense then takes the field and tries to score. If the offense scores a field goal (worth 3 points) or fails to score entirely, the other team is given a chance to either tie the game with a field goal of their own or win the game with a touchdown (worth 6 points). Should the game be tied after each team has had a possession, the next score will win the game regardless of what type of score it is.
If the first receiving team scores a touchdown, the game is over and the opposite team does not get a chance to score.
Teams have to be a lot more precautious with their play calling in overtime since a turnover in your own half of the field can lead to an easy game-winning field goal. Field positioning is often more important than taking risks.
An additional 10-minute period is added to the game for overtime. If there is no score or the game is still tied after the 10 minutes have expired, the game ends in a tie.
In playoff games that cannot end in a tie, another 10-minute period will be played after a small intermission until a team has won the game. No playoff game has ever gone past double overtime.