The ale horn is a drinking utility, used as a fancy and extravagant substitute for the regular mug. It comes in both a regular and a golden variety, the latter of which has no additional benefits other than to show off your wealth.
Despite of it's name, the ale horn can hold any kind of drink, not only ale. To fill it, right click on a barrel of the drink that you want to fill the horn with, and then either place it down on a solid block, or drink it by holding down right click.
Right clicking on a placed ale horn will make you drink it instantly. Also, you can right-click an empty vessel with a full one in your hand, or vice versa to transport the fluids to and from each other.
If you don't like to steal ale horns from buildings, and would rather hunt some wild animals, then you can craft your own. You need to hunt down a rhino, a Kine of Araw or an aurochs. Gemsbok horns won't work, probably because they are not curved enough.