Overtime in the NFL has been rewritten more often than almost any other rule in the sport, because the league kept running into the same complaint: a coin toss should not decide a playoff game.
The regular season
If the score is level at the end of the fourth quarter, the game goes to a single overtime period of ten minutes. There is a coin toss to decide who receives.
Crucially, both teams are now guaranteed a possession. Under the old sudden-death arrangement, a team could win the toss, drive the length of the field, score a touchdown and win without the opposition offence ever taking the field. That is no longer possible.
Once both teams have had the ball, the game reverts to sudden death: the next score of any kind wins. If the ten minutes expire with the score still level, the game is a tie. Ties are rare but entirely legal in the regular season, and they count as half a win and half a loss in the standings.
The postseason
A playoff game cannot end in a tie, so the structure changes. Overtime periods are fifteen minutes rather than ten, both teams are guaranteed a possession, and if the score remains level, another period is played. And another, if necessary. Play continues until somebody is ahead at the end of a period or scores in sudden death after both possessions have been used.
There is a break between overtime periods, and teams change ends.
The situations people get wrong
A defensive score during the first possession does end the game immediately. If the receiving team throws an interception that is returned for a touchdown, that is a walk-off, because the opposition has scored on its own possession by definition.
A safety also ends the game immediately, whenever it happens.
If the team that receives first kicks a field goal, the other team gets the ball and needs at least a field goal to continue. Score a touchdown and it wins outright. Fail to score and it loses.
Two-point conversions are live in overtime, and a team trailing by two after a touchdown may go for the win rather than tie the game.
Why it kept changing
Every version of the rule was tested by an actual playoff game that felt wrong. Sudden death produced games decided before one offence touched the ball. The field-goal compromise produced games where a touchdown on the opening drive still ended it. The current guaranteed-possession model is the answer to the fairness objection, at the cost of longer games.
The short version
Regular season: ten minutes, both teams get the ball, sudden death after that, ties possible. Postseason: fifteen-minute periods, both teams get the ball, repeat until somebody wins.