I take a large pot and quarter apples - skin, seeds and all. I find that Ida Reds or similarly very red skinned apples are best. Because I cook them with the skins - it gives a beautiful reddish tinge to the sauce that I love. Cover the apples with water and bring to a boil. Reduce heat and continue to cook until the apples are broken down and resemble a wet applesauce. Most of the water should have been cooked off - sometimes you may have to add some extra water if things get too dry.
I have a piece of equipment that my mother used to make applesauce. I have no idea the actual use for this - or other uses for this gear. I always called it a "colander". It has a metal stand - a metal "colander" that is triangular in shape, and a wooden tool that is used to move the ingredients around inside the colander and release the sauce out of the wholes. (I used this for the grape juice also - although I lined the colander with several layers of cheesecloth as I only wanted the juice - no pulp).
Once the apples are cooked down - pour it into the colander (which has been placed over a large bowl) and start moving the wooden dowel around and removing the sauce. You will need to use a spatula to take the skins away fromt he sides to allow for more sauce to get through. Process all of the apples and then toss the skins, seeds and cores. What is left in your bowl is applesauce.
I fill jars and process them in a hot water bath for 20 minutes. Remove from bath and tighten lids.
This equipment can be found at http://www.lehmans.com/ . This store is known for carrying "non-electric" equipment for the Amish community. It was a small local store until the year 2000 when the Y2K scare and people started researching this store. As a result it has become quite the tourist attraction and carries many items that are very helpful. When people come to visit - it is on the list of places to take them.