Sudden death is a term used in hockey when it's overtime and the first goal scored will end the game.
If the teams are tied up at the end of a regular hockey match, the two teams will go into overtime. Each side will put out three players and one goaltender on the ice for a five minute overtime period. Sometimes, teams will elect to bring out an additional player instead of the goaltender. No matter what, there are less people on the ice so it is easier to score during this time. If one team scores, the game is over and whoever scored wins.
If neither team scores, then there will be a shootout. Three players will go one on one against the opposing team's goaltender. Whichever team scores the most goals wins. If they tie again, there will be a sudden death shootout where the winner will be the first team to have a player score when the opposing team can't.
For the regular season, overtime works as explained above. However, during the Stanley Cup Playoffs, overtime is a little different. Instead of having a five minute overtime period, the overtime periods are 20 minutes. Both teams have five players that go head to head and the first team to score wins. There is no shootout so if nobody scores, then they keep playing 20 minute periods until one team scores.
In the National Hockey League, each team will receive an extra point to benefit their league standings if they go into overtime. This is regardless of if they win or not, but the team that does win will get an extra point. If a team decides to get rid of their goaltender in exchange for an extra attack player and they lose in overtime, they will get no extra points.
A lot of other popular sports have a similar sudden death format in their overtimes. American football uses the sudden death format so that the first team to score a field goal or touchdown wins the game. Golf also has a sudden death component. If two golfers are tied, they will play an extra hole until a winner is decided. Other sports with some type of sudden death component are rugby, baseball/softball, fencing, and wrestling.
The longest overtime in NHL history happened during the 1936 Stanley Cup Playoffs with the Detroit Red Wings and Montreal Maroons. The two teams were in the sixth period of overtime when Mud Bruneteau of the Detroit Red Wings finally scored. There were only 3.5 minutes left of the sixth overtime period. This means they almost played two extra full games in overtime. The Detroit Red Wings went on to win the Stanley Cup that year.