A bye, in sports and other competitive activities, most commonly refers to the practice of allowing a player or team to advance to the next round of a playoff tournament without playing. This is generally the result of having a number of entrants in the competition that is not a power of two (i.e., not 4, 8, 16, 32, etc.); any such tournament must eventually through elimination arrive at an odd number of participants at some point, thus necessitating the bye.
In a typical use of the term "bye", the NFL rewards two division winners from each of the two conferences which possess the best regular season record with a bye in the playoffs. This is necessitated by the current 12-team playoff structure. Eight is the next lowest power of 2, so four teams must be eliminated in the first week to get down to eight teams. The NFL is the only one of the "Big Four" North American sports leagues to offer playoff byes.
https://en.wikipedia.org/wiki/Bye_week
Bye (1) -- In sporting use, a variant of by (prep). Originally in cricket, "a run scored on a ball that is missed by the wicket-keeper" (1746); hence, in other sports, "position of one who is left without a competitor when the rest have drawn pairs" (1883).
http://www.etymonline.com/index.php?search=bye&searchmode=none
