Some acoustic and popcorn ceiling material contains asbestos, according to the Environmental Protection Agency (EPA). The use of asbestos in building materials was banned in the USA in 1977 because it can cause cancer and serious respiratory problems, so if your home was built before then, you must test the popcorn material for asbestos before removing it. You cannot tell whether your ceiling contains asbestos fibers merely by looking at it. Contact your regional EPA office to find a testing lab. They will also be able to tell you if it's legal to remove asbestos in your state without a license and can advise you on proper and safe procedures.
You will need a roll of heavy-duty plastic, several large trash bags, a dust mask and eye protection for removing popcorn texture. Even when it doesn't contain anything harmful like asbestos, it's probably nothing you want to breath in. Clear the room of everything possible, including window treatments. Cover what you can't remove with plastic, and protect the floor with drop cloths. If you have to wet the ceiling to remove the popcorn, put plastic sheeting underneath the drop cloths.
First, try simply scraping the texture off the ceiling with a wide, stiff bladed scraper. If you're lucky, it won't require wetting first. If it doesn't come off easily, you'll need to wet it down. Use a spray bottle or a pump garden sprayer for even faster results. The trick to getting it off easily when wet is to saturate the popcorn texture thoroughly. Note that you will have to use a lot more water if the ceilings have been painted because the paint will prevent water from penetrating easily.
Underneath the texture, you'll probably find that the ceilings were not finished smoothly--most builders skipped the final application of joint compound and sanding, knowing the ceilings were going to be covered by a thick layer of popcorn texture. You'll need to apply another coat of joint compound over the joints, followed by a light sanding, for smooth ceilings. Once that's done, you're ready to prime and paint.