Drifts are the most non-conventional of any of Need For Speed: Underground's challenges. For one thing, there aren't any other cars on the track; for another, your lap times don't matter one whit. Instead, you'll be required to rack up points by Powersliding your way around a small course that's been specially slicked-up just to reduce traction. Drifts are frustrating before you learn the ground rules, but when you figure out what to do, they're a welcome respite from the breakneck speeds of the other types of challenges.
Powerslides are generally a bad thing in Need for Speed: Underground, since they often indicate that you've lost control of your vehicle due to taking a turn too fast or hitting a patch of dirt. In Drift mode, however, your main goal is to hit Powerslides, preferably in as flashy a manner as possible. As your loading screen will occasionally remind you, the two keys to scoring high in Drift challenges are speed and the angle of your slide. Achieving one of these keys is easy; combining them both without ramming into walls is quite difficult.
Points
First, though, a note on scoring. You should keep in mind that the point totals for Drift challenges are distinct from Style Points (although you will rack up huge Style Point bonuses when you master Drifting). Your score increases as you maintain a Powerslide, and will keep counting up until you either stop moving or stop Powersliding. Of course, a single Powerslide will be maintainable for only a finite period of time, due to the interference of the walls. Thus, you'll need to chain your drifts together to achieve truly outrageous point totals (details below!). If you do hit a wall while sliding, all the points that you've racked up during that Powerslide will be lost.
To make things more interesting, there's a Bonus Zone that rings the racecourse; it's the area with parallel lines that goes all the way around the track. If you get two or more wheels into this zone while Powersliding, you'll get bonus points. Your slides inside of this zone will see their points multiplied by the number on the left side of your screen, which itself is increased as you score more points. So, higher points lead to a larger multiplier, which in turn leads to more points, and so on. Of course, the multiplier is set back to its default setting if you hit a wall. You must also be above a set speed limit for the multiplier to kick in, so keep your car moving.
Technique
Now that you know a little about scoring, it's time to take your car out to the track and rack up monster points. The most critical aspect of getting big scores is to perform slides at a high speed; you can control slides better if you're puttering around at 20 mph, but your slide scores will increase very slowly, and you won't get the benefit of the score multiplier. So go fast, and prosper. (Note that the very first Drift track you encounter isn't very conducive to getting high speeds, so don't worry if you can't apply this right away; the later Drift tracks are a bit more open.)
Second, you'll need to coax your car into traveling almost perpendicular from the plane of the front end to get more points during a slide. In other words, if you're sliding for monster points, your car should wind up traveling almost sideways, especially in turns. While the slick roads will help you out here, you'll still want to get things going by hitting your emergency brake, then giving your wheel a tap into the corner you're going through. This will let your car's rear end fishtail out, resulting in more points than a slide with a less drastic angle. Of course, your car will be harder to control, and if you hold an extreme-angle slide for too long, you'll usually find yourself ramming into a wall.
Walls are the end point for any slide, actually, unless you learn how to get your car out of slides as well as you can get it into them. The trick is to not hold down the throttle, but rather to feather it or, if you're on a console controller, rapidly tap it, while moving your wheel back and forth to prevent your car from either going into a wall or ending the slide. This is easier to do than to describe in words; a little practice will be required before you can do this consistently. Don't be discouraged by wall impacts; Drift races are short and restarted easily enough if you fall behind the leaders.
Of course, for truly monstrous point totals, you'll need to chain your drifts together. In a perfect situation, such as a wide straightaway, you can begin a Powerslide off to one side of your car, then shift it to the other side to extend the length of your slide, and repeat the process to score one massive trick total. This requires a bit of touch, of course; you'll need to feel when your car is reaching the point of no return in a Powerslide, beyond which you can't prevent it from hitting a wall. Before it reaches the breaking point, you'll need to let go of the gas completely, jerk your wheel in the opposite direction, then re-apply the gas. When done properly, your car should reverse its course and start Powersliding in the opposite direction, thus giving you an uninterrupted influx of points. If you spend too much time readjusting, however, and wind up traveling forward for more than a second or so, the chain will be broken.
Beyond the basic mechanics, though, it's critical that you hit your Bonus Zones if you wish to win an event. The middle of the track is obviously the safest area to slide in, since the risk of hitting a wall is relatively low, but you'll rake in the points in the Bonus Zones.
Strategies
Since your Style Points aren't tied to your progress in Underground mode, and you can rack up massive amounts of Style Points in Drift competitions, you can unlock most of the vinyls in the game by running around Drift tracks in Quick Race mode. If you use a car with a Reputation multiplier of three or higher, you can easily get over 50,000 style points per Drift event; try setting the lap number to 10 to get even more Style Points in a single challenge.
Keep in mind that there are significant differences in the way that rear-wheel drive and front-wheel drive cars perform; FWD cars are going to be more difficult to put into Powerslides and will thus be a bit harder to get high scores with. Since trading in cars is essentially free, you may want to trade in an FWD car for a RWD vehicle, like the Miata, before heading off to the Drift tracks. All of your upgrades will carry over to the new car, and you'll find that the rebate and costs of trading from your normal car to the RWD car and back balance out, thus eliminating any kind of penalty to your bank.
There's no cheating allowed! If you attempt to turn your car around, you'll be automatically reset facing forward, and if you attempt to reverse your car to try a section again, you won't accumulate any points until you reach a part of the track that you haven't yet hit on that lap.
Tip: Although unequipping performance parts is a tactic that works best in the racing portions of Need For Speed: Underground, it can also help you with Drifting, especially towards the end of Undergound mode. A car that's fully tricked out with level 3 parts will be a monster on the Drag track, but the extra acceleration and top speed will only help you ram into walls quicker during Drifts. If you find yourself having a hard time controlling your car, try taking everything down to level 2. You should find it easier to avoid wall contact and switch your drifts from side to side.