Its an opportunity to take something like an intro level college course over the summer while in middle school with gifted kids your own age. You need to qualify by taking the SAT and scoring above some threshold and you can only take courses for subjects you scored well in (a high math score doens't qualify you for the language arts courses). A huge number of prominent people in all fields went to CTY.

I went in 7th grade to take Algebra over 3 weeks. It was pretty fun and remember being sort of blown away at the final presentations by what other groups had accomplished in their time. My group was pretty boring and just did did problems out of a textbook...

Honestly, my biggest memories from that were spending time with other kids (namely girls), especially since this is like peak puberty age. Its also at an age where its easy to make friends living in a dorm for a month.

I went to a similar course studying abroad, and it wasn't nearly as good. I took an overview course on physics in high school. It wasn't a math based course, so it just devolved in to pop science. And it was at an older age where people are much more judgemental and I remember thinking I'd prefer avoiding a good number of people for various reasons. I never had that while at CTY.