When you are driving there are certain times when there is a danger of losing traction. Two of these times are trying to stop on wet roads and making sharp turns at high speeds (worse again on wet roads).
To understand this answer we need a bit of physics. Every surface has two coefficients of friction, a static coefficient and a dynamic one, these must be tested for as I'm not aware of any way to accurately tell just by looking at something what its friction coefficients are. The static coefficient is a measure of how much force it takes to move from rest, the dynamic coefficient is a measure of how much force it takes to continue moving. Typically, the way a car moves is that it's tires push against the static coefficient of friction in such a way as to not overcome it, but use its resistance to generate forward motion of the car.
The problem is that the momentum of the car is strong enough to overcome the static friction (which is what drives your car) and allow your car to move without being driving by the opposing forces of static friction and the rotation of your tires (the movement equation here is the rotation of the tires versus the dynamic friction of the road). This is what allows drifting. Obviously drifting is a bit more complicated than this as there is lateral motion in addition to forward motion, but essentially the idea is the same.
Typically drifting is accomplished by over steering. You go into a turn fast and crank the wheel hard instead of lightly. Without over steering your momentum would carry you straight or at only a slight angle, but with over steering your tires loose traction (that is they overcome the static friction of the road). This actually slows you down just a bit through the turn, when you slow sufficiently, provided everything goes correctly your tires should slow enough that they can no longer overcome to the static friction of the road and return to grip.
The Wikipedia article linked above has this to say about the cars typically used:
Usually, drift cars are light to moderate weight rear-wheel-drive coupes and sedans over a large range of power levels. There have also been AWD rally cars that have been converted to RWD.
And goes on to summarize several other features that are important including the drive train, suspension and tires. I find the tires to be the most interesting part as they are the piece of the car that contacts the road and primarily affect the physics of the manuever:
Competitive drifters often run DOT-approved tires closer to racing tires, which is permitted, with the exception of some major championships including D1GP which only permits commercially available tires that are approved by them. Professional drifting has come to a point where the maximum amount of tire grip is necessary to be competitive in terms of sustaining speed, and stability in a drift.
Grassroot level Japanese cars with low horsepower quite often have different tires on the front and back. The tires with more grip are used up front and harder compounds in the rear to be able to spin the rear wheels in a higher gear while still being able to maintain a relatively moderate speed in a drift.
A lot of grip will resist drifting, but will also allow you to catch better on the road surface allowing you to regain control quickly after a slide through a turn.
As far as the stunt in the movie, I haven't seen it, but IF you has a sufficiently high rate of speed, a sufficiently round and smooth course, I'd have to say that you could slide all the way around a circle, however it would be very difficult as if you lost the slide you'd be in big trouble.
Final note: Drifting is incredibly dangerous as it involves losing control of your vehicle, it may also damage your vehicle, please do not try this at home.