What is the best way to remove popcorn from the ceiling?
My wife and I just finished doing this last month. Here's the process we took:

Before you start, check to make sure you don't have asbestos in the popcorn. If your house is newer construction, you don't have to worry about this, but if it's older, you may want to scrape a bit off the ceiling and send it to a lab for testing. If it comes back positive, I would have professionals handle it. Otherwise:


Cover everything you don't want to get gross and disgusting.
Turn off breaker for the lights and remove fixtures.
Use a spray bottle or a garden sprayer to wet 5' by 5' sections of ceiling with water. Be careful not to oversaturate it and damage the drywall.
Scrape off popcorn with a 6" or 12" taping knife, whichever you find easier to handle.
Sand any spots you missed that are still rough. (For the most part, everything comes off smooth.)
Apply mud liberally to any gouges you made. Most spots I had to go over were at the seams of the drywall.
Sand again.
Repeat 6 and 7 if you're OCD like me. If you want to get everything super smooth, shine a flashlight or work light parallel with the ceiling to find raised/gouged areas.
Wipe down the ceiling to get rid of any sanding dust.
Prime
Paint with 2 coats of ceiling paint.
Throw away drop cloths
Lie on the floor and stare at your new smooth ceilings :)


One bit of advice: You're going to make A LOT of dust, and it will get all over everything in your house that isn't covered. We did the whole process before we moved in, so we only had to wipe down the walls and floors.

After you're all done, you'll probably want to change your air filter as well.