The Fulbright Scholarship is a prestigious award typically given to students planning to pursue independent studies or to teach English abroad after graduation. Some grad students and professors are also awarded Fulbrights, as research and postdoc teaching grants, respectively.
You can apply for a Fulbright via the Fellowships Office. Your chances are improved if you want to go somewhere less conventional than England, France, or Germany. It's reputedly easiest to get a Fulbright to Paraguay.
13 Columbia graduates received a Fulbright for the 2007-2008 year of 71 who applied. That places Columbia tied for 6th among Ivy league schools, with nearly half the number of awards as Yale (27) and Brown (25). In addition, 3 Barnard graduates were also awardees for 07-08.
For a list of Columbia Fulbright recipients see Category:Fulbright Scholars.