This French term literally translates to hazelnut butter, because the butter is heated right until the moment you see specs of brown bits and it exudes the fragrant aroma of roasted hazelnuts. It is most often referred to as brown butter in recipes.
The process of making beurre noisette is heating and melting the butter on low heat which allows the butter to separate into butterfat and milk solids. Initially the milk solids will look like white foam floating on top of the butter fat. As the water from the butter begins to evaporate, the milk solids sink toward the bottom of the pan. Once all of the water evaporates, you can actually hear the loud crackling of the butterfat, begin to see the little bits of brown butter turn light amber in color at the bottom of the pan and start to smell the rich, distinct nutty aroma. Make certain to swirl the pan often so that the brown butter does not settle to the bottom and burn. You must be careful however, as it is easy to burn the butter, so you must monitor it closely and remove the pan from the heat as soon as turns a lighter shade of brown.