In the NFL, Overtime conditions state that if the team to possess the ball first score a Touchdown on their first possession, then they win the game. If they score a Field Goal, or fail to score, then the other team has a chance to possess the ball.
Now let's look at the following examples:
Team A has won the toss in overtime and has elected to receive the kickoff. Team B kicks the ball and it is returned by Team A. During the return, the runner fumbles the ball and it is recovered by a member of Team B.
Does Team B need to score a Touchdown or a Field Goal to win the game? As they are the first team to run a down.
It is 4th down for Team A, on the first possession of overtime, and they decide to Punt. Team A punts the ball down the field. A fair-catch is called, but the returner from Team B does not cleanly collect the ball, and muffs it. It is then recovered by Team A.
Does Team A still need to score a Touchdown to win the game, or do they now only require a Field Goal?