MADELINE: You doing the summer school thing too?
RORY: Uh, yeah.
In North America, schools and universities can offer academic programs which take place during the summer vacation. In high school, students may enrol in classes for credit which can improve their grade point average or be included on their academic record. Summer school can either make up for credits lost through absence or failure, accelerate progress, or lighten the course load for the year.
Summer school usually lasts for 3-5 weeks, although Louise complains about spending “the summer” at Chilton; she’s probably exaggerating. You don’t normally have to wear a school uniform to summer school, so Chilton is very unusual in expecting this. In fact, most schools make money by attracting students from outside their school, especially international students, so a uniform would actually by impractical.
Rory missed the first few weeks of the academic year at Chilton, and at first struggled to keep up, so it makes sense for her to enrol in summer school.