It is difficult to define what a tiebreaker is without sounding redundant. It breaks ties. It's in the name. And it is not hard to see why they exist: in an elimination tournament you need an exact number of winners, and an exact number of losers, be it a single or multiple elimination tournament. Not just in situations where the tournament is time-limited - Even a rematch is a sort of tiebreaker. It is less clear to me why they exist outside of that specific context - In casual play, it doesn't matter who wins and who loses, while in non-elimination tournament structures I'd argue that breaking ties distorts the ability to accurately rank competitors which is the reason tournaments exist.
As an example of this, consider the following results of a simple round-robin between four teams:
Team A Team B Team C Team D
Team A --- W L D(L)
Team B L --- L D(L)
Team C W W --- D(L)
Team D D(w) D(W) D(W) ---If ties are broken (brackets for results of the tiebreaker), that would lead to the following results table:
1) Team D - 3
2) Team C - 2
3) Team A - 1
4) Team B - 0
If the ties are left unbroken (awarding a half-point for a tie):
1) Team C - 2.5
=2) Team D - 1.5
=2) Team A - 1.5
4) Team B - 0.5
While if we go for something that ranks ties below half a win, such as the 3 for a win, 1 for a draw system seen in the likes of Association Football
1) Team C - 7
2) Team A - 4
3) Team D - 3
4) Team B - 1
In the first example, the team that won wasn't able to secure a definitive win, while the team that was able to win two matches in regulation - The team that I would argue played better during the tournament - came second. Some league structures have tiebreaks while trying to untangle this problem; for example, the NHL offers 2 points for a win and 1 for an overtime loss.
But what makes for a good tiebreak when they're needed?
Ideally, a tiebreak will reward the same skills as regular play. The obvious conclusion? The best tiebreak is to replay the game, but this is rarely practical. Outside of that, shorter, miniaturized versions of the game (e.g. overtime), who was closest to increasing their score further, etc, can be used.
Similarly, the worst tiebreakers will have absolutely nothing to do with the game itself - a coinflip, planting a tree and seeing who's grows tallest in five years (As in Arboretum although that, at least, has some social good attached to it), who can throw the game off of a roof first (As in Nothing Personal). At best, these will be amusing but utterly useless in situations where they're needed. At worst, they result in election results being determined by pure chance.
Another way you can have a bad tiebreak is if the rules of the tiebreak lead to unnecessary player exhaustion. While this can occur in tabletop or video games, the obvious examples are Basketball and Baseball's unlimited overtime periods/extra innings.
A prime offender for bad tiebreaks in competitive environments is Association Football, which has historically awful tiebreaks, and every attempt to correct this seems to make them worse. It is a typically low scoring game, where goalless draws are not uncommon. In elimination tournaments played on strict schedules, it has a 30 minute period of extra time, divided into two 15 minute halves, with a penalty shootout afterwards if extra time fails to break the tie.
Penalties are fairly universally acknowledged as being a very bad way of breaking ties in association football - It has very little relation to 99% of the rest of the game while putting unnecessary pressure on individuals in what is otherwise a team sport. They're about as close to the coinflip option for tiebreaks as you can get without literally involving things outside of the skillset required for the game. Meanwhile, due to how low scoring association football is, 30 minutes of extra time when played to completion isn't likely to break the tie.
As an attempt to mitigate this, the 'golden goal' rule started to be rolled out in 1993 and was fully in place by 1996, which made extra time first to score in 30 minutes wins instead of leader after 30 minutes wins, meaning that any game that wasn't a goalless draw after extra time would have a definitive winner. Unfortunately, if you concede a goal, you lose encourages hyper-defensive play, and a team focusing on hyper-defensive becomes very difficult to score against in association football. This increased the proportion of games that went to penalties, the opposite of the desired effect. An attempt to fix this - the 'silver goal' rule - was implemented around 2002 to 2004, where a team with a lead after 15 minutes of overtime will win, without playing out the remaining 15 minutes. It worked so well that since 2004 the tiebreak is back to playing out the full 30 minutes, which while ineffective is less bad than attempts to improve it proved.