Math Circles are usually a fun way for middle or high school (or sometimes even younger) students to get together with a mentor to do some fun, exploratory math together. Math Circles, at least for me, really emphasize the idea that math is about discovering, finding patterns, asking questions, and going where the questions lead us -- as opposed to rote learning and memorizing algorithms, which school sometimes makes us believe math is.
Teachers Math Circles, such as the one in Boston I was part of for a few years, bring together middle or high school math teachers. Together we work on open-ended math problems, make some discoveries and have a lot of fun along the way! We also discuss relevance of these problems to the school curriculum, and other issues related to teaching and learning math. Have a look at my "presentations" page to see the few sessions I led myself.