Perhaps the most famous number ever worn by an athlete, this is the number worn by Michael Jordan in high school, at University of North Carolina, and, of course, with the Chicago Bulls.
According to Professor Pi, Jordan picked 23, because his brother had #45, and he figured he was about half as good.
This number was used in a number of Nike's evil marketing schemes, and was often found on on the shoes, that bore his name, the Air Jordans.
When Michael "retired" the first time and went to play baseball, with the Birmingham Barons a AA affiliate of the Chicago White Sox, he also wore #23.
Upon his return to the NBA, he wore #45 (his brother's number again) because of some strange rule (I never got the full lowdown on that). Either that or he was trying to just get more jerseys sold. However, in the second season of his return, he went back to wearing #23.
Upon his second retirement, there was some talk of retiring the number league-wide like the NHL did for Wayne Gretzky's #99. However, there were too many players wearing it, most in tribute to Jordan, so he asked that they not do it.
23 is also the name of Jordan's Chicago restaurant.
UPDATE: Jordan wore #23 for the Washington Wizards after he returned, again.