Budgies are usually referred to as "tamed" when they have been trained to step onto a human finger on cue and sit quietly. This normally takes about two weeks but varies with the individual bird.
Taming is a gradual process of winning the bird's trust and is more easily accomplished with young birds than older ones. Steps in taming vary somewhat from trainer to trainer but can usually be summarized as 1) getting the budgie accustomed to human company in the same room, 2) getting the budgie accustomed to a human hand in its cage, 3) getting the budgie to step up onto a human finger offered as a perch while still within the cage and 4) getting the budgie to stay quietly on a finger outside the cage. Patience and frequent rewards are the keys to the taming process.