A player who has "honors" is the player who hits first from the tee box. On the first tee, honors can be determined randomly or by any means desired.
Thereafter, the player with the lowest score on the preceding hole gets honors on the next tee. For example, a player who made par goes before bogey, who goes before double-bogey and so on. In the case of ties, order of hitting from the previous tee carries over.
A golfer who cards the lowest score on a hole is said to "have the honors" or "have honors" on the following tee box.