Back in those days we cooked the cored apples with the skins still on, running the resulting hot mash through a manual food mill. With all the sprays and pesticides used now, it's better to peel the apples first, but only after washing them well with soap and water and rinsing them carefully.
Scrub well, peel, quarter, and core 6 to 8 tart cooking apples (Gravensteins are best, if you can get them. Granny Smiths are good, too)
Place the apple pieces in a large saucepan with just enough water to barely cover the bottom of the pan
Bring the water to a boil, turn down the heat, and cover the pan. Don't leave the area! Just hang around the kitchen doing sudoku and check on the pan from time to time. You want the apples soft, not scorched.
Now you have some choices. Once your apples are cooked, you need to turn them into sauce, although some people like the chunky stuff. You could use an old fashioned hand-operated food mill like the one pictured here, or something similar. You could use a food processor. Because it's handy and because I don't have the old mill my mom and I used, I use one of the gadgets that came with my KitchenAid mixer.
Whichever you choose, run your apple mixture through and put the resulting sauce back in the pan. Turn the heat to low and start adding sugar, a little bit at a time, until your sauce is the desired sweetness. Apples vary and so do tastes, so I'll let you figure this part out without any measurements from me. Keep the pan over the fire until the sugar has dissolved.
Add a little vanilla and you're done.
If you decide to make a really big batch--and why not? You're messing up your kitchen anyway--this freezes well. You can freeze it in small plastic containers or, to save space, in 1-quart ziplock bags. Just spoon the sauce into the bags, flatten out the sauce and zip the bag slowly to let out all the extra air. These flattened bags can be "filed" upright in an oblong container after they have been frozen solid.
To make a spiced sauce: Add some cinnamon, freshly ground nutmeg (you don't use any other kind, do you?), and possibly some cardamom and/or mace. I add these spices to the sauce after it has been thawed and warmed.