Our pipe coming out of the wall was leaking a very fine stream of water. We haven't had time to fix it. I don't know what that bumpy stuff on the ceiling is called but it's all peeling off. I know the pipe needs to be fixed first, but how should we fix the ceiling? Is it possible to apply it back on the ceiling? None of it has actually fallen off yet.
Most likely it is some sort of drywall texturing applied by hand or with a spray gun. If you're asking if it can be re-attached to the ceiling - I don't think so. If you're decent at DIY jobs, and you can identify the texture, you may be able to apply new texturing yourself. I've been able to get decent results, but not perfect using a product like this : http://www.homedepot.com/buy/building-materials-ceilings-attics/homax-10-oz-orange-peel-spray-texture-3-pack-119845.html
Keep in mind that it will probably require some surface prep and new paint.
What's up there depends on the age of the home and how much the contractor cared when they spec'ed it. Most likely, the ceiling is some form of gypsum board that is then plastered, textured and painted. The result of water leakage on gypsum board is usually that the entire board starts to sag under its own weight; alternately, the outer layer of paper, along with any spackle, texture and paint, will peel away with the damp.
Unfortunately, drywall repair == drywall replacement. You have to cut away the damaged portion, fit and screw a new piece of drywall into that hole (using rust-resistant screws as this area already has a reputation for moisture), then tape, spackle, seal, texture, prime and paint.
On top of that, many types of ceiling texture such as "popcorn" ceilings are simply impossible to blend in with when you do a repair like this; tiny differences in the application of the texture along with the sudden stop/start to the old texture around the patch will make repairs obvious. To really do the job right, you have to re-texture the entire ceiling by scraping off the old stuff, smoothing the surface with sandpaper or plaster, then reapplying texture, priming and painting.
If you're going to do all this anyway, I would recommend replacing the entire piece of drywall that has been leaked on with a more water-tolerant product like backer board (DuRock, Hardee-Backer, etc). This product is normally used in areas where water and/or heat can be a problem, and are durable enough to get soaked without crumbling apart unlike gypsum board. These products are generally a different thickness than drywall so you'll need to level the ceiling with lath strips of the proper supplemental thickness on the joists. After that, installation should proceed much like with drywall (you may need to smooth the surface with more plaster; backer board is commonly used as underlayment for tile and so often has a mesh that grips thinset).