Rival events in classic Bokujou Monogatari games would showcase marriage candidates and the rivals for their affection. These rivals would be trying to woo the same candidates as you and could beat you to wedded bliss if their rival event chain reached its conclusion before you made your move. Modern-day versions of the series do not contain rival events because surveyed players in Japan don't like them.
The original Friends of Mineral Town game contains four rival events that would result in the event couples getting married. These rival events trigger based on your love point level with the marriage candidate. The criteria for triggering rival events were measured by the candidate's lack of affection towards you as well as the current year in the game. Candidates needed to have a minimum heart level or a lower heart color for the rival event. Basically, these rival events were the opposite of romantic heart events and would trigger at low affection levels. If a candidate had a black heart color, all of that candidate's rival events could trigger after completing the required number of in-game years.
In an attempt to balance old and new, the remake of Friends of Mineral Town only contains the first two classic rival events for the marriage candidate couples. These two events could be triggered in year 1 in the GBA games. The missing two events, the green rival event (year 2) and the orange rival event (year 4), have been left out of this game. Omitting the rival marriage system also results in several random events left out of SoS:FoMT because the events required the rival couple to be married.
Triggering the rival events does not impact your love points or friendship points with the candidates. Rival events can be seen after you get married, except if you happen to marry the other half of the rival couple, of course. Nobody will try to woo your sweetie away from you.
Brandon and Jennifer are characters added to the GBA remake and do not have rival events.
Because of the low heart level required to trigger the rival events, once one-half of the rival couple reaches a higher heart level than what the rival event requires, the rival event won't trigger at all. You will have to lower the candidate's heart level to comply with the rival event trigger requirements. This can be tricky for the rival events between Popuri and Kai, as Popuri could be at a purple heart color by the time Kai shows up in Mineral Town during Summer. The black rival event will not trigger if she is above a black heart color. Give Popuri terrible gifts, such as Branch or Lumber, to reduce her friendship points to below 10,000 LP.
Seeing these events is not mandatory. Nothing in the game will change based on whather you see or don't see the rival events. You can see more than one rival event a day.