[noun] + は + いくつ + ありますか [noun] + を + いくつ + [verb] [noun (person)] + は + (お)いくつ + ですか
いくつ asks "how many" about things you count with the ~つ numbers: ひとつ一つ, ふたつ二つ, みっつ三つ. It goes where the number would go in the answer, and the question ends in か.
いくつ goes with the ~つ numbers, which count objects that have no counter of their own. People, flat things, long things and the rest have their own counter, and you ask with なん何 plus that counter.
いくつ also asks how old someone is. With お in front, おいくつ is the polite way to ask an adult; なん何さい歳 is more direct.
The answer counts years with とし歳. A small child's age can also be given with the ~つ numbers.