A bye is a game mechanic used in tournaments.
Because tournaments use a single-elimination bracket, tournaments are easy to run only if they have a number of players which is a power of 2: i.e. 8, 16, 32, 64, 128, 256, etc. Having this exact number of players ensures that in each round, all players have somebody to compete against. It is unlikely that this exact number of players will enter, so "byes" are used to fill holes in the bracket.
A player gets a bye when there is simply no opponent for him or her to play that round. Getting a bye is considered fortunate for a player; he or she is guaranteed to advance to the next round. Sometimes many byes are granted in a particular round. The aim of the byes is to have the number of players be equal to a power of 2 in the next round.
For example, in the bracket diagram shown above, Yubaba and Whitevelvet both received byes (note that the image is not of the complete tournament bracket). Tournaments are always set up so as to create a number of players equal to a power of 2 in the second round.
In a seeded by rating tournament, byes are given to the highest-seeded players, who hold the highest puzzle standing in the respective puzzle when the tournament begins.
Getting a bye and advancing to the second round of a tournament does not put players "in the money" for cascading tournaments. Players must win at least one match to be able to win money from tournaments, subject to the settings of the tournament.