The specific requirements to qualify onto the team vary from year to year, and by the type of gymnastics
In 2008, both the men’s and women’s artistic teams will select their 6-member teams by committee. The committee will weigh the performance of each gymnast at Nationals and Olympic Trials, his/her strengths on each apparatus, and his/her past experience.
In rhythmic gymnastics, the US did not qualify any competitors to the Olympics, so unfortunately no gymnasts will be sent to Beijing. (The top 20 athletes from the 2007 World Championships qualified, and no US athletes placed that high.)
In trampoline, the two athletes (one man and one woman) are selected by the total points earned in four different competitions throughout the year.
In order to be considered, all candidates must be United States citizens and must have qualified to the elite level
See the complicated rules for each team selection:
Women's Artistic Gymnastics
Men's Artistic Gymnastics