How does Korean chili powder differ from "US" chili powder?
Korean chilli is a little different as it has a slight smoky flavour, in addition to being slightly sweet and also quite hot. 

The actual name of the chilli use in kimchi and for that matter, most Korean dishes is gochugaru (고추가루). It comes in a variety of preparations, typically, finely ground, flakes and a paste. 

You should be able to find this in most good supermarkets or an Asian store. If you can't get this, you can still use a good quality chilli powder or possibly, a paste.