http://www.stephanietolan.com/is_it_a_cheetah.htm is my favorite article of all time about identifying gifted kids.
For myself... my son is perfectly-normal-boy. But (probably because he's ADHD) he falls quite solidly in the "gifted range". Reading fluently by 3, chapterbooks like the Jack & Annie and the Black Stallion by 4, algebra by 7... these are all very normal / nothing exciting things in my family. ((We've had 3 honest to god geniuses in my family... so reading by 3 but only in English and not War & Peace, algebra at 7 but not advanced calculus/applied mathmatics, etc., so forth & so on... just isn't extraordinary. It's normal fun childhood stuff. My husband, for example *is* a genius in patterns he absorbs them like a sponge. It took him a year to master everything for a Comp Sci degree -but he got degreed in dead languages-, he's been signed and preforming music since a teenager -about 2 years after he started playing-, he speaks more languages than I can count. Music, languages, programming, anything with a pattern just makes sense to him without having to think about it... my uncle had his doctorate by 22... my cousin was building working computers by 5 and viruses by 7 and is the definition of 1337 h4x04...my grandfather was world reknowned blah blah blah... those are the "preformers" btw that are mentioned in the above article. And they are almost inhuman. But they're genius range, not gifted range... which are two different playing fields.)).
But even though my son is perfectly normal to ME, and in my family, nothing "special" we ran into major problems in school. The reason one doesn't hear the phrase "gifted adults" (although one does hear about brilliant adults) is that as adults we get to choose our paths. The brilliant diplomat or film director or mechanic or astrophysicist or athlete or dancer or teacher is following a *chosen* path. Kids, however, are all put into a homogenizing place known as school. And in most schools what one can do/work on is strictly defined by age.
Preschool was GRAND. A montessori school, kiddo was able to work on what he chose for how long and in as much breadth or depth as he liked. He'd come home showing off the cool new thing he learned in math, or talking excitedly about the "Mom! Did you know the brain has a big CRACK in it! It's called the longitudinal fishy!" (longitudinal fissure) In areas he *adored* he soared as far and as fast as he wanted, or he could stay on one thing and do it to BITS (I have over 50 hand drawn maps of the world from when he was 4, and that's after tossing out most of them). In areas he didn't much like he wasn't forced to do them for hours on end. But as soon as we went into public K... whoops!. Misery. (Sort of... kiddo is an extrovert, as long as he's surrounded by people he can keep himself amused). Academically, he was bored out of his mind, which is a double curse being ADHD. I was rather horrified to find out that the school's curriculum wouldn't "catch up" to what he was doing in preschool until the 3rd grade! And that was the *lower* end of what he was doing academically. Mathwise (kiddo has always been a "numbers guy"), they wouldn't catch up until grade 6 or 7. (Now part of this is a "geography" thing. In some states/schools those numbers would be much lower, but he was at one of the "top" schools in our state.)
A bare minimum of 3 years before he learned anything NEW? 6 or 7 years before he got to work on the things he LOVED?
Even as an adult that's a heinously long time. For a child... that's literally a lifetime.
His K teacher was one of those you dream about. Amazing woman who would simultaneously challenge the advanced kids (breaking school and district rules and getting reprimanded left and right for it... advantage of being tenured and over 30 years experience she *could* choose to be on the outs with the district and they couldn't sack her) while making the ESL kids or kids who didn't even know their colors like they were a million bucks.
She strongly recommended that we get kiddo out of the public school system as fast as possible and either send him to the private gifted school or homeschool him. The gifted school was *amazing* and would have been perfect for our "2e" kiddo (kids grouped by age but working at their own level, breaks every hour for recess, running/skipping/cartwheels/stopping to play with a rock or a slug ***encouraged*** between classes, off the wall questions receiving applause instead of "We're not talking about that right now"... utterly, utterly amazing)... but with a pricetag of 15k a year, even though he was accepted... we chose to homeschool.
My son is perfectly normal (sigh, for a massive extrovert, it's not 'talking to strangers' that we have to discourage... it's inviting them and their kids and their neighbors and their 2nd cousin twice removed HOME... kiddo is a natural host)... but he amazes me every day. He hates handwriting. He'll grump about having to do anything he doesn't want to. "But Mooo-oooooom...." and trying to define him by grade is almost impossible (so we define it chronologically). What he works on in school spans 7 or 8 grade levels. We're doing highschool physics in science and and 1st grade work in English at the same time. History we can't even quantify as to grade level (he'll tell you all about the Old Kingdom in Egypt for 2 hours without taking a breath, but not be able to write a paragraph -yet!- about it. We've spent over a year on Ancient Greece -personal gripe-) We have a *lot* of perfectionist and adhd issues we deal with on a daily basis. I often feel like I'm raising a young Robin Williams (did Robin's mother survive his childhood??? Is there hope?) We have an absolute blast.
I don't think of my son as gifted, but he falls quite solidly into the range.
For the best info on gifted kids... do check out http://www.hoagiesgifted.org/