To quaff is to gulp. You’re in a desert; you haven’t had water in days; you’re wicked thirsty. Then you find an oasis! You lean over and quaff the water.
If you’re in a room with your favorite foods, you might stuff the food into your mouth. Quaffing is the same fun, but with beverages instead of foods. But just like with stuffing food, quaffing means you drink a little too much. Often, quaffing refers to alcoholic drinks, but it can mean any beverage in general, like milkshakes. Quaff a milkshake and you’ll get a headache, but you’ll have fun doing it.