Well that's a slightly complex question, as "Burgundy" came to refer to the entire territorial complex ruled by the duke of Burgundy in the 15th century. Burgundy proper was incorporated to France in 1477, after the death of Charles the Bold at the Battle of Nancy. The territory was held as an apanage, in that it was only to be passed down to legitimate male sons of a French king, in this case the Valois Duke of Burgundy was descended from king John II (the "Good"). Louis XI seized the Duchy of Burgundy, now referred to as Burgundy (as well as Picardy in northern France), in the chaos following Charles the Bold's death without a male heir.

However, it is important to note that the rest of the House of Burgundy's territories -- the Low Countries (what's now Belgium and much of the Netherlands) as well as the Country of Burgundy (now known as France Comté) passed to the house of Habsburg. Franche Comté was conquered by Louis XIV in 1674.

Sources: Joseph Calmette's The Golden Age of Burgundy, Richard Vaughan's Charles the Bold.

Edit: I'd like to point out that although I've mentioned in this post and the one below that the Duchy of Burgundy was an apanage, it must be said that in fact the Duchy of Burgundy was in fact an independent principality John II inherited and gave to his son as an apanage. This is slightly different than it actually being an apanage as it was never formally incorporated into the royal domain as say Alençon or Anjou were. This is actually a big legal difference I forgot to point out and further undercuts the juridicial basis of Louis XI's annexation of the duchy.

Hope this helps. I know a lot about the period if you'd like further detail or have other questions.