It can be repainted. You should purchase a special roller brush to paint it and use paint sparingly. The ceiling does not to be saturated or you will find yourself having to repair it which is a difficult process unless you can save the pieces that come apart and put them back by painting the pieces on the flat side and placing the pieces carefully back in place.
It is not necessary to remove, but I would think about doing it. Popcorn ceilings are no longer "in" as they say. Most people want them taken down when they purchase a home. If your planning to sell in the future, now is the time to do that extra work.
Once painted removal becomes even harder as you must also scrape not only the popcorn, but the new paint covering it.
But if you want to keep it, Rent a small spray unit, mask the walls with paper tape and plastic and spray the ceiling. You can also use a roller, but it often does not cover really well as you need to get all around the nooks and crannies if there is such a thing to cover and make the ceiling look good again. The roller should be a long nap type.
Lastly if you decide to remove the popcorn, do so carefully. Depending upon the age of the material used. It could contain asbestos materials. It can be tested fairly easy. First take a small amount and using a lighter see if it melts when the heat hits it. If that happens it is more then likely Styrofoam beads like you find in bean bag chair. This is a very common material in newer ceilings installed around the early 80's If it does not melt, then take a sample put it in a plastic bag and send it to any one of the many labs you can find on line for testing. It should cost around $35 for a test.
Next once you know what your dealing with, do a search on this site at the top of the page for popcorn removal. There has been lots of great advice for doing this. Not a pretty job but not that hard either.
Once painted it is indeed harder...check out this article I wrote on removal