Bafög is need-based assistance given to German high school and university students until they complete their first degree. The need-based calculation contains a bewildering variety of factors that include the parents’ income and the student’s expenses. High school students don’t have to pay it back, and university students have to pay half back, without interest. The word comes from the abbreviation for the law that established the scholarship, the Bundesausbildungsförderungsgesetz [German education/training assistance act].
Update on 27 May 2014: After weeks of discussion about spending more on education, the federal government has announced it will start paying 100% of Bafög, rather than the current 65% federal money, 35% state money. The states will have to invest the 35%, ~1.17 billion euros each year, in schools and universities, though Wolfgang Schäuble said there are no control mechanisms in place to enforce this. Bafög is to be reformed again in 2016/17.
Update on 21 Jul 2014: The federal government announced they will be increasing Bafög by 7%, the Elternfreibetrag [?] by 7% —which will increase the number of students qualifying to receive Bafög by >110,000 university and school students—and the rent stipend to 250 euros/month. The increases will go into effect in the winter semester 2016/2017. The last time Bafög was increased was in 2010.