Brown butter has an earthy, nutty flavor that really shines through in autumnal flavors. (Just try pumpkin ravioli with brown butter and sage.) I use brown butter in my Heeeere's Johnnycake! cookies, and I'm going to use it in my soon-to-be posted pumpkin cookie recipe.
I use salted butter when I make my cookies (see here), but you should use unsalted butter when you make brown butter. Salted butter will make brown butter that tastes very salty. Melt the butter over medium heat and stir often. The butter will melt and get really foamy. Keep stirring.
The foam will disappear, and the butter will start to have clear bubbles. Keep stirring. The butter will start foaming again and turning brown, and the solids in the butter will start settling to the bottom of the pan. Remove the pan from the heat. Voilà! You've got brown butter.