As the USCF Handbook puts it, there is no perfect tiebreak system; each has its faults. Here are the tiebreak systems which SwissSys supports, along with a brief description of how they are calculated.
Of the two median tiebreaks, this is the more standard now. It evaluates the strength of a player's opposition by summing the final scores of his or her opponents and then discarding either the highest of these scores, the lowest, or both, depending on the tied player's score. For players who tie with even scores (that is, their number of wins and losses is the same), both high and low are discarded. For tied players with plus scores, only the lowest is discarded, and for players with minus scores only the highest is discarded. For tournaments of nine or more rounds, the top two scores are discarded (or the bottom two scores, or all four, as determined by the same even/plus/minus criteria above). These scores are adjusted for any unplayed games, which count a half point each. If the player involved in the tie has any unplayed games, they count as opponents with adjusted scores of 0,
Basic median (Harkness median)
This works just like the modified median above, but highs and lows are discarded regardless of the tied player's score. Generally the modified median is preferred to this nowadays.
This is just like the Median except that no opponents' scores are discarded. Popular with tournaments of only a few rounds.
This is easy to calculate by hand, and has been popular for that reason. To get this value just add up the cumulative (running) score for each round. The theory is that players who win their games in the early rounds (and therefore end up with higher cumulative scores than players with the same score who win later rounds) have had to face tougher opposition throughout the tournament.
Cumulative scores of opposition
This uses the cumulative scores calculated as above, but for the tied players' opponents rather than for the tied players themselves.
This is just like the cumulative, but discards the first X rounds, where X starts at 1, and rises until the tie is broken. You must set the number of rounds to discard whenever you select this tiebreak. However, unlike the other systems available in SwissSys, you can select this tiebreak more than once, specifying a different number of rounds to discard each time.
This system rewards aggressive play by scoring 4 tiebreak points for a win, 2 for a draw, 1 for a loss, and 0 for an unplayed game. For players with equal scores in a tournament, the one with fewer draws will have the better tiebreak score.
Performance of opposition
This method averages the performance ratings of the players' opposition.
This averages the ratings of the player's opponents.
Drop-low average opposition
This is just like the average opposition, but it drops the low value before calculating the average.
Head-to-head can be an undependable tiebreak system, and there are three reasons for this, First, there is no way to determine a clear winner in cases where A beats B, B beats C, and C beats A. Second, in ties involving two players, a draw does nothing to resolve the tie. Third, in ties involving more than two players, it is hard to decide a ranking when some players have lost to players in the same scoregroup, but others have not even played against anyone in the scoregroup.
In spite of this, head-to-head is still a popular tiebreak in some circles. SwissSys uses an arbitrary 100 point basis for players, then decreases that value by a point for every game lost against any one of the other tied players.
This straightforward method gives the edge to players who have had to contend with playing black more.
Extra for black (BAP system)
This awards tie break points according to the following system: Win as black = 3, win as white = 2, draw as black = 1, all other results = 0.
Round robin (Sonneborn-Berger)
This system is most often used for round robin events. For each player in the tie, add the final scores of all the opponents the player defeated and half the final scores of all the opponents with whom the player drew.
Game points (Fixed-roster)
This adds up the individual games won by a team, as opposed to that team's match points.
This is another fixed-roster team tiebreak. For each round, the final score of the opposing team is multiplied by the number of points scored against that team. This is recommended over the game point tiebreak listed above because it compensates for lopsided victories against weak teams.
Another straightforward method which, like the Kashdan system rewards aggressive play.