The 15 and 17 indicate the width of the rim, 622 is the bead seat diameter of the rim (also called 700c).

In general, any 700c tire will fit as long as it is not narrower than the rim. A tire much wider than the rim (like more then 2x) is not ideal but will work. Aside from that, the only limit is how big a tire will fit inside your frame/fork/brakes.