Ghee (pronounced GEE with a hard G) is the pure butterfat left over after the milk solids and water are removed from butter. It's used widely in Indian cooking, and the word ghee is the Hindi word for fat. Ghee might as well be a synonym for clarified butter, although there is one slight difference.
Like clarified butter, ghee is made by melting butter, cooking off the water and separating the clear, golden butter fat from the milk solids.
The only difference is that in some traditions, ghee is simmered for a little while, thus browning the milk solids and adding a slightly nutty flavor to the finished product. Not all ghee recipes necessarily specify the browning of the milk solids, however, so for all practical purposes ghee is clarified butter with an Indian name.
Ghee is better for high-heat cooking than butter since it has a smoke point of between 450°F and 475°F, as compared with about 350°F for ordinary butter.
Another advantage of ghee is that it has a longer shelf life than ordinary butter and, when stored in an airtight container, can be kept at room temperature.