Gifted children ask a lot of questions - they are curious about how things work, and the why's and wherefore's of the world. They are often individualists, even thought by their peers to be weirdos. They are often quiet and thoughtful, and often have unusually good vocabularies, and seem to reason things out better than some of their less-endowed friends of the same age. That's my impression of the gifted children I know.
One or two people answering this question have said that it doesn't matter whether children are gifted, etc. etc., but, whilst that might be the case, it wasn't what was asked,as the question was. "how do you know?" and not "does it matter".
I always tell clever children that they are fortunate enough to have a special gift of intelligence above the average, and that they should use their gift for the benefit of mankind.