AI and AIS lenses, are almost always fully compatible with pro-level DSLRs, meaning D200, D1/1h/1x, D2h/x/hs/xs.
Fully compatible means: manual focus (of course, all AI/AIS lenses are), and metering in aperture preferred or manual mode. Shutter preferred and Program are not supported. (Literally they are, although if you select them, you actually get aperture preferred mode anyway.)
The only exceptions might be some perspective control lenses, but even at that I know that most of the 28/f4 PC and 35/f2.8 PC lenses are fine.
As Brian said, they will work fine in "A" or "M" mode on your D200.
When using Ai(S) lenses on the D200 (as on the D2 generation bodies) you can tell your camera which lens it uses. In the camera menu, select the option "non-CPU lens" and program the focal length and max. aperture of your lens. The camera will then display the correct aperture and the lens info will show correctly in the EXIF file.
As other said, the only thing you'll loose is the AF. However, setting the proper focus will be easy. You'll just need to frame, half-press the shutter while you rotate the focus ring,(till the focus green circle shows up perfectly still, in the left corner of the viewfinder), meaning you achieved proper focus.
On the other hand, some of the Ai/Ais still available even used have fine optical perfomances, sometimes better than the AF.
All well said and now a word of caution: Do NOT mount a non AI lens to a modern DSL without modifying the Mount, you can damage the mirror on your new DSL! However it is an easy fix many articles aroundon how to do it or send it off to John White who will do it for you for a small sum for you.
The cool thing is that pretty much any old Nikon F mount lesn can easily be modified to work (exception some special lenses like 8mm fisheyes etc. that requires a raised mirror to operate, special caution with them!) YOu cen learn much more about the old lenses at Bjorn Rorsletts site
My favorite AI'd lens is an old 58mm F1.4 from 1959, great color and pretty sharp when stopped down. Another AI lens to reckon with is the 75-150mm series E zoom, sharp as you know what and very cheap!!!