How do you paint a ceiling that is badly stained with nicotine?

Patches. Lots of them.

There's a paint product called Kilz that should work well. I used it to paint over a cement wall stained with soot from a furnace, with good results. It's available at any of the popular home improvement stores. There are two types of Kilz stainblockers -- an alcohol-based product that is highly volatile and should not be used without an organic-vapour-cartridge respirator, and a water-based product that, while marginally less effective, is considerably less toxic. Either variety will accept a top coat well (ceilings are usually finished with a low-cost, non-washable flat latex paint. Ask your paint-store employee).

prime it with "KILZ" primer and then finish coat with your selected latex paint color