Burgundy is nowadays a region in France, but in medieval times it was a quite powerful state-like entity between France and (future) Germany. From time to time it stretched from the Netherlands to Provence. It's borders fluctuated constantly. Counties and duchies were conquered, bought and inherited, acquired through weddings and lost in wars. It was a multicultural and loosely controlled common market area.
Important cities in Burgundy were Brügge, Antwerpen, Ghent, Amsterdam, Lille, Arras, Brussels, Dijon.
Burgundy was one of the most enlightened and civilised region in Europe at that time, somewhat comparable to the Renaissance city-states in Northern Italy.
The Pope hoped Burgundy to help the emperor of the Holy Roman Empire to defeat the Turks, who had conquered Constantinople in 1453 and threatened Europe from the east. In exchange for this help Charles the Bold demanded sovereignty for Burgundy.
The emperor of the Holy Roman Empire, Frederick III, chickened out at the last moment; he feared that Charles the Bold would demand the crown of the emperor for himself. Charles was enraged by this treachery and went to war with his neighbours. He was killed in a battlefield in 1477. His daughter Marie married Maximilian Habsburg, the son of the emperor. This was pretty much the end of Burgundy.
(This information was from an essay book I'm currently reading)
Here's more information:
http://en.wikipedia.org/wiki/Burgundy
http://en.wikipedia.org/wiki/Duke_of_Burgundy
http://en.wikipedia.org/wiki/Dukes_of_Burgundy_family_tree
Here are some nice maps of the lands of Burgundy:
http://employees.oneonta.edu/farberas/arth/arth214_folder/burgundian_expansion.html
http://employees.oneonta.edu/farberas/arth/arth214_folder/burgundy_intro.html
The possible flag of Burgundy:
http://www.fotw.us/flags/fr-bg.html
And of course there is this:
http://en.wikipedia.org/wiki/Lotharingia
What if Burgundy continued to exist? What if Frederick didn't chicken out? What if Charles the Bold became the founder of a sovereign kingdom of Burgundy? What if the multilingual and multicultural nation-state of Burgundy existed even today as a kind of "buffer state" between France and Germany (replacing, say, the entire OTL Belgium, parts of the Netherlands, parts of Germany/France, stretching down to the borders of Switzerland)?
How would European (and the world) history from 15th century onward be affected? Especially the relationship between France and Germany? Would there be the kind of antagonism and warfare between these countries as in OTL before 1945? Would the history of Europe be more peaceful or even more violent with Burgundy hanging around? Would the continued existence of Burgundy be even possible from the 15th century up until 20th/21st century?