Curing- it is a process that involves allowing rock to sit in water to allow dead organic material to rot off. It's sole purpose is to prepare rock for aquarium use. Curing is necessary when using live rock that has been out of the water for a little bit (such as in shipping) or when reusing dead rock.
Let me clarify these terms..
live rock- rock that comes from the ocean or another tank that has organisms and bacteria living on it. When it is shipped or out of the water for a bit some of the organisms will die, this is referred to as "die-off"
dead rock- used to be live rock however it has been removed from water long enough that the vast majority if not all of the life on it died and dried up.
Dry rock- rock that has not been in a tank yet and was pulled from the ground, has no (detectable) organic matter on it.
If you were place dead rock or live rock that has had die off on it into an aquarium, the die off and dead organic matter will begin to rot which will add nutrients to your water and cause an ammonia spike. This is where curing comes in, curing is basically allowing your rock to sit in water long enough that all of the dead organic matter on it rots off and all you are left with is live rock. Normally this is done in a trash can because the rotting material causes an odor that most wouldn't appreciate in their home. In addition if you cure rock in your tank you will likely be faced with high nutrient levels as a result of the decaying matter, basically you start on the back foot if you cure rock in your DT. It is doable and many have done it, however most do not consider the easiest way to do it. The amount of time it takes to cure rock depends solely on the amount of organic matter that is present on and in the rock. If rocks came out of a tank covered in algae and then dried out, it will take much longer to cure than live rock that was shipped overnight.
Cycling- is a process of building up a biological filter in your aquarium that will be able to process fish waste through the nitrogen cycle. It involves placing your rock into your display tank and feeding bacteria with a source of ammonia, as the bacteria multiply the ammonia levels will be converted through the nitrogen cycle into less harmful nitrates.
So in conclusion- Curing prepares your rock for the aquarium, Cycling prepares your aquarium for fish. They can be done at the same time however there can be complications to the process (notably the smell). Live rock that has not been out of water for any length of time and Dry rock require no curing process. This is one of the factors leading to growth of dry rock popularity in the hobby.
Hope that helps