This is the name of a forest south-east of Paris and the site of a magnificent hunting lodge converted into a great palace by the French monarch Francis I (1494 - 1547).
In terms of painting, art historians refer to the two 'schools' of Fontainebleau. The first is the description given to the Mannerist work carried out in the chateau by Italian artists Francis employed there, most notably Niccolo dell'Abate, Rosso and Primaticcio. The second school is the name applied to artists of the later 16th century working mainly for King Henry IV, who developed a style which owed much to the work of earlier painters.
See also the entry for Barbizon School.