You pretty much just let the squeezings sit. That at least is what early Americans did. Given time (ten days or so), yeasts on the skin of the apple go to work consuming the sugars and giving off alcohol. But that’s just one possible way to go.
In some cider-making traditions the fermentation is carried out before the cider is even squeezed. Back in Devon, cider makers form the apple pulp into large, dense cloth-covered blocks that they refer to as “cheese”. The blocks are left to sit for several days until the pulp ferments, at which point they stack the blocks under a large press and squeeze the whole works into a barrel. This is pretty much what “scrumpy” is. Filtered and refined it can be turned into clear hard cider, of the type you find in bars and liquor stores. Just how much alcohol does cider have in it? Anywhere between 3 and 9%, about the same as beer which averages about 5.5%.
The next step up the ladder in terms of alcohol content is apple whiskey or “applejack”, which was a fairly big thing in colonial America. Applejack was mostly a poor-man’s liquor, made by simply allowing hard cider to freeze in the winter, then taking off the ice. But while this sort of distillation was a fairly effective way of increasing alcohol content, it didn’t necessarily produce either great flavor or consistent results. Which is why more sophisticated applejack makers employed steam distillation of the type used for corn whiskey. Nowadays, unfortunately, true applejack is very hard — if not impossible — to find. Distillers simply don’t consider it worth the time and effort to make it. The cloyingly sweet stuff you find in stores is usually just apple brandy diluted with apple juice.
It is of course always possible to make one’s own out of a large quantity of homemade hard cider. But then I would never do that. That would be illegal.