Other forms: buddings
Something that is described as budding is growing brand new buds. A budding rose bush is getting ready to burst into bloom. If you are in a budding romance, it is also just about to burst into a new phase of excitement. Ooh la la.
When a tree or flower is budding, it's showing signs of the branches and blossoms that will soon appear. The same is true of a person who's described as budding — if you're a budding filmmaker, you're new at making movies but already show great promise. Budding most likely comes from the Old French boter, "push forward or thrust."