The Congress-Bundestag Youth Exchange Scholarship is a full scholarship from the United States Congress and the German Bundestag. It is specifically for high school students to study and live for a year in Germany. CBYX was founded in 1983 to strengthen ties between the US and Germany, and over 23,000 students have participated. There is also a fellowship program for college students, called CBYX for Young Professionals. 250 high school scholarships are awarded each year, with 50 available for each of the five regions: YFU, ASSE, AFS, FLAG, and CIEE. Visit the CBYX website to find out what district you live in.
In addition to CBYX, there are two other US Government sponsored scholarships:
YES Abroad is a scholarship for eleven countries with significant Muslim populations: Bosnia and Herzegovina, Ghana, India, Indonesia, Macedonia, Malaysia, Morocco, the Philippines, Senegal, Thailand, and Turkey. YES only offers year programs. There are 50 scholarships available.
NSLI for Youth is a scholarship for learning less common languages, and offers summer and year programs. This year's languages are Arabic, Mandarin Chinese, Hindi, Korean, Persian (Tajiki), Russian, and Turkish. Since the program focuses mainly on language, it includes multiple countries that speak the same languages, like Jordan, Morocco, China, Taiwan, Latvia, Russia, etc.
Along with these government scholarships, I really recommend the Rotary Youth Exchange Program. Rotary is very cheap compared to larger organizations (normally $4,000-$8,000). They have exchange districts all over the world, including "less popular" countries like Iceland, Thailand, Slovakia, Kazakhstan, The Philippines, and South Africa. Get in touch with your local Rotary club for more information.