Being bored in school is also a characteristic of students that have to deal with boring teachers. :) Oh, and DON'T SAY THAT AT THE TEACHER CONFERENCE! heh heh! (I'm not picking on teachers - it's not easy for ANYONE to keep a bunch of young children mentally engaged for hours every day.)
For an idea of "characteristics of the gifted child" see:
http://www.hoagiesgifted.org/characteristics.htm
Hoagies has lots and LOTS of information giftedness, various gifted programs (most of which are $$$, but not all), advice for parents and kids, and lists of recommended gifts (like challenging games or toys) and links to cool (FREE) websites.
The term "gifted" is generally used loosely - many schools have *some* kind of program for gifted (I think "TAG program" for "talented and gifted" is a term used in Georgia public schools) but it means different things in different places. Maybe it's the top 10%, 5%, or 1% based on some objective test. Or maybe there's a more subjective criteria, such as when a child is referred to the program by a teacher who believes the child shows particular creativity or aptitude in some area.
Then there's "gifted," "highly gifted," "exceptionally gifted," and "profoundly gifted" (think Mozart).
Unfortunately for gifted children, many states have had to cut the budget for gifted programs as an unintended result of "No Child Left Behind," because in some schools there is so much focus on bringing the struggling students up to speed that the kids at the other end of the spectrum don't get enough stimulation. (I don't want to imply that that is wrong, just that is has had side effects, at least in some schools, since educational resources are limited.)
Similarly, much of the high-stakes testing intended to increase "accountability" has had a high opportunity cost in other areas, but I digress.
Keep in mind that many very bright children act out in school and can be misdiagnosed as having ADHD and my be inappropriately medicated. And there are also many gifted children who having learning disabilities, known as "twice exceptional". Giftedness frequently accompanies dyslexia, Asbergers, ADHD, etc. If your child is having enough problems at school that you're one day considering medication, THAT would definitely be a good time to have a psychological evaluation. They are usually pretty expensive, though. I know a woman who has had her 3 children tested by (I think) Stephen Garber and Nancy Hatcher, (2 different practices) both in Atlanta or Marietta area. And I think it set her back around $2,500 - $4,000 per kid - it varied for each of her children. And it's kind of an ongoing thing, since they each have their gifts and their issues.
In general, the best things to do for gifted children holds true for *all* children, so read up on what gifted programs offer or do differently and try to do the same kinds of things with your kids. Keep reading aloud to them, even after they can read on their own. Give them lots of opportunities to experience different things and help them find their passions. Give them lots of outlets for creativity and don't over-schedule. And when they find something they're really interested in, help them learn more, obsess and really get INTO whatever it is, chemistry or painting or fencing. You know - help them nurture their passions, even if it seems odd to you.