The Fulbright Program is the United States’ largest international cultural and educational exchange and has several programs under its general umbrella. It's called Fulbright because it was founded by Senator J. William Fulbright and it dates to 1946--immediately Post-WWII, when the United States realized that the soft diplomacy of international cultural exchange at the person-to-person level could do much for the US globally. Since its founding more than 70 years ago, the program has evolved and grown to include the Fulbright-Hays (which funds, among other things, doctoral dissertation research) and the ETA (English Teaching Assistantship), which provides young Americans with the opportunity to teach English around the world, and their host countries the opportunity to have native US-dialect English speakers embedded in schools teaching students ranging from elementary through university levels.
Fulbright funds around eight thousand people all over the world every year, including both foreign nationals to come to the US and US citizens to conduct independent research, teach English, undertake group-based projects and provide professional expertise in practically every country globally with which the US has diplomatic relationships.
The Fulbright Program you initially might think of when you think of a “Fulbright” if you’re in the US is the Fulbright US Student Program (the Fulbright-IIE) that provides grants for US citizens to conduct research and other projects in more than 140 countries. It's a program with broad possibilities--during any given year, grantees might be writing a play, composing music, or doing research in a scientific lab, in addition to teaching English in cities ranging from capitals to far-flung villages.
If you've finished or are about to finish a BA but have not completed a PhD, and you're a US citizen, being selected for the Fulbright US Student Program is a life-changing opportunity.