A single-elimination event is one in which teams (or individuals) are dropped from the competition after a single loss. Some bad quizbowl formats utilize single-elimination playoffs after an initial pool or bracket, which eliminates a contender from playing once they lose a match in those playoff rounds.
The single-elimination bracket works pretty much exactly like the March Madness playoff bracket in NCAA Division I basketball. Teams are seeded based on previous results (in quizbowl, this is typically the results of the morning's bracket or pool play). A typical single-elimination bracket features some even number of teams, usually a power of 2 (like 2, 4, 8 or 16), which are paired so that the highest-seeded team plays the lowest-seeded team, the second-highest plays the second-lowest, and so on.
After the first game of the single-elimination playoffs, the losers of each game are eliminated from further play in the tournament. This leaves half the number of original teams to continue playing. They will progress one step further in the bracket and play a team who also just won a game. The losers of those matches would also be eliminated until there are only 2 contenders to vie for the title. (It is not unusual for there to be a 3rd place match held between the teams that just lost to the championship contenders.)
Single-elimination playoffs are considered bad quizbowl. In limited cases, only a single-elimination playoff may work for a tournament (due to time, staffing capability, field size or packet availability), but it is to be avoided if possible.
The biggest reason that good quizbowl advocates generally avoid single-elimination playoffs is that it limits a team's exposure to the content of further packets in the tournament. Teams which play more matches are exposed to the information in those questions in a quizbowl-specific context. Also, teams who attend a tournament have generally paid an entry fee to be able to play the event. Teams who leave because they have been eliminated have paid the same fee as teams which stay until the finals are complete, and have thus paid for questions that they were unable to hear.
Another reason that single-elimination is avoided by good quizbowl is that the results of one game have a proportionally high impact on the team's continual play in the tournament. A team which has one 'bad game', gets a moderator they have trouble understanding, experiences buzzer issues or is mistakenly denied a protest could suffer elimination based on that one outcome alone. Especially in tournaments with less quality control on moderators or packets, this can be a major issue. Using a format other than single-elimination allows teams to at least try and recover from those problems.