Apple mash is what is left over after you juice the apples. I use a juicer which collects the pulp. You could also use an apple press. There's some skin and pulp and a few chunks of apples in it. I take the pulp, put it in a gallon jar, cover with water and about a cup of live, apple cider vinegar to it and let it set on the counter, or in the cupboard for a few months, then strain. The result is a light bodied, apple cider vinegar that is great for salad dressing and fish. If you want it stronger, it needs to set for longer to evaporate and condense.
(you can also do this with just apple peels.)
Boiled cider is just that. It is left on the stove on low, until it condenses into a syrup. Very yummy! Great on pancakes and french toast.