Well, you ask a question which neither medical nor psychological science can fully answer.
Originally IQ was for finding out who might be retarded with regards to school learning. Binet was asked to create a test by which the French school system could identify those kids that would not be able to learn in a normal setting. Binet took the approach that people learn more the older they get. Therefore to him IQ was a function of mental age (how much kids new; arrived at by answering common knowledge questions in a test) divided by chronological age. He established a baseline of comparison by testing thousands of children of each age. Thus, if you are 10 and answer as many questions as the average for 10 year olds, you have a 100 IQ and are therefore normal. The standard deviation on the IQ scale was likewise directly carried over from the SD of the testing he had performed. There are obviously many MANY flaws with this system, which is why Stanford-Binet testing has fallen out of use.
Wechsler changed things by saying there was a difference between knowledge and intelligence. He further broke with tradition by establishing different testing for preschool, school, and adult testing. By splitting IQ into general information and pure reasoning sections he was able to establish a slightly better distinction between knowledge and intelligence and one less susceptible to variance by culture or upbringing. It was still not the whole answer however.
Gardners work has been nothing short of revolutionary, by identifying individual areas of intellect based not so much on acquired knowledge but on potential and ease of acquisition in 8 key areas. I believe his work more than any other explains the phenomenon you're describing...people who are brilliant in one area, but slow as molasses in another. People can be musical prodigies, and not have a clue how to spell, or how to deal with others, or how to do a handspring. Other people can natural dance and paint, but not be able to solve simple math problems. Gardners multiple intelligence theory provides answers here.
Quickly gaining in popularity is Sternberg, who declares that intelligence is really just adaptability in 3 key areas; componential (book smarts), experiential (people smarts), and contextual (life smarts). The biggest problem with Sternbergs theories is their subjectiveness. It's almost impossible to really quantify these things, especiall across cultural borders. Even so, it's a simple concept and one which provides some insights.
All of these methods have some things in common and some differences, but none completely address the core question of 'what is intelligence'. Biology can't completely attribute it. Philosophy provides no solid answers. It's enigmatic at best.
Speaking as someone who's spent a lifetime in programs for 'exceptional intelligence' I have decided that while there are definately some core distinctions between someone average and someone 4 or 5 standard deviations above normal, IQ as it's commonly understood is not an accurate measure of much. The curve doesn't accurately represent differences between retarded, normal, exceptional, and truly brilliant. Nothing discusses the sociological impact of these different score ranges, and believe me, IQ is much more about what the world does to us than what we do to or for the world. Grades have nothing to do with IQ, although they vaguely have some relation to knowledge. Income/careers have little correlation, although extremely high or low IQ's are equally detrimental to modern workplace dynamics. All of this has led me to pursue my degree in education, in hopes of developing more balance between IQ and Personality and Schooling and Society. It's really a fascinating field of study if you're interested.
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