Aging lets the rigormortis leave the muscle tissue, it also lets the muscles tissue break down so it's more tender. Most rigor is gone in 4 hours for poultry after 24 hours it's completely gone and some muscle tissue breakdown starts.
Generally Ice and plain water is used to keep the meat cooler then the 38 to 40 degrees that most refrigerators keep meat, because the FDA recommends storing poultry at 36 degrees which is lower then most refrigerators will chill. The disadvantage is that you need to completely change the water several times during the resting and aging process.
Water and brine in the refrigerator is used to preserve the meat some and have some extra muscle tissue break down, this is done 24 to 48 hours. The disadvantage is the extra salt in the meat. Brine can actually speed up the process of rigor leaving the muscles tissue.
Plain water in the refrigerator keep air from getting to the meat and drying it out. again 24 to 48 hours. But it's a good idea to change the water at least once, more then once is a good idea
No water and refrigerator, when using this method the meat needs a barrier to keep the air from drying out meat.
Which method you actually use is up to you they all have some disadvantages and advantages. We use the water and ice method, we freeze water in milk jugs for a couple a weeks before we butcher. We use an old apartment freezer that that is about 4 x 4 x4. since it has no drain in it we have a portable sump pump that remove most of the water.