Short answer: it's a gathering of people who play the Native American Flute for fun, personal enrichment and fellowship. It's nothing more complicated than that.
It's not about being "Native American", although most folks who attend a flute circle cherish the rich cultural background of the Native American Nations. It's not about being "New Age", although some folks find a stong identity in New Age philosophy. It is about the music, and everyone in attendence wants to hear Native American flute music or improve their skills with the Native American flute.
Each flute circle, and each event for that matter, has it's own personality. I've been to flute circle events where it was nonstop spontaneous playing for three hours, and I've been to other events where all we did was share stories and talk about flutes and flutemakers for three hours. I've been to events where everyone was a beginner and other events where everyone was a seasoned musician of some sort or another. I've been to other flute circles where I never touched a flute and played percussion all afternoon while everyone else jammed along on the flute or other instruments.
Regardless, the most important thing in my opinion is to have fun and be spontaneous at any flute circle event. I love discovering new melodies when playing a duet with somebody who I have never jammed with before. I equally love seeing somebody "light up" with excitement when they learn a new technique or find some new source of inspiration which they've never had before.
I've made some wonderful lifelong friends through my experiences with the Northern California Flute circle. I hope that you find your experience with your local flute circle/clan to be a similar experience. Just remember that flute circles are a grass roots organization. The flute circle is nothing more and nothing less than what everyone together contributes to the organization. Whether you are leading an event, hosting an event or attending an event, everyone contributes to the experience. I encourage you to take the time to welcome new faces and include everyone in the experience. Invite somebody new to a circle. We always have a meal as part of the flute circle, because nothing builds a community like sharing a meal together.
If you would like to find a flute circle in your area, you can either look online at the NCFC website (for Northern California and surrounding areas) or the INAFA website for a listing of worldwide flute circles.