Cobbler: Cobblers have been a part of American cooking since at least the early 19th century, though in their early days they were more like proper pies than the deconstructed dish they are today. While always a combination of fruit and pastry, cobblers usually feature one of two kinds of toppings—either an arrangement of biscuits or shortbreads (both the cut and "drop" styles) or one made from a looser, cakier batter that's poured on top of the fruit before being baked.
Crisp: Even simpler than a cobbler, a crisp is much as it sounds: a dish made mostly from fruit, cooked until jammy, and topped a rich "crispy" topping that is typically made from a mixture of butter, sugar, flour, and sometimes oats.
Crumble: This is where it gets tricky. For many folks, the terms crumble and crisp are more or less interchangeable. Both feature fruit covered with a crispy, crunchy layer of sugar and butter. But generally speaking, crumbles can be a little more elaborate than crisps—the topping may be slightly thicker and clumpier, or include nuts as well as oats, as well as spices like cinnamon, ginger, or cardamom.
Buckle: More like a conventional cake, in a buckle the fruit and batter is swirled or spooned together, so that the mixture puffs and "buckles" around the fruit during baking. Though sometimes plain, they are often finished with a buttery-sugar streusel topping.