Techniques for inducing drift

Beginner techniques
These techniques don't use weight transition, so are typically the first thing the novice drifter learns. However they are still used by the most experienced drifters, and require skill to execute properly. These techniques aim to induce a loss of traction on the rear wheels, either by locking the wheel (e-brake drift), or using enough power from the engine to break the traction force (power-oversteer and clutch kick).
Hand-brake or Emergency brake drift - While the clutch is depressed, the hand-brake is pulled to induce rear traction loss. As soon as traction is lost, the driver releases the clutch, depresses the accelerator, and countersteers. This technique is used heavily in drift competitions to drift large corners, or to trim the car's line mid-drift.
Power oversteer or Powerslide - This drift is performed when entering a corner at full throttle to produce heavy oversteer through the turn. The excess power causes the drive wheels to lose traction in a RWD or AWD car. This is the most typical drifting technique for all-wheel drive cars.
Shift lock (compression slide) - Initiated by downshifting (usually from third to second or fourth to third, and using a very fast shift) instead of braking, without rev-matching, causing the drive wheels to lock momentarily. Helpful for very tight corners, allowing the driver to approach the corner at a slower speed and lower revs, while allowing quick acceleration when exiting the corner. This technique can be very damaging to the engine if mis-used as the ECU is unable to rev limit when the engine is oversped by the rear wheels. Premature downshifters are called "Rod Stretchers".
Clutch kick - This is done by "kicking" the clutch (pushing in, then out, usually more than one time in a drift for adjustment in a very fast manner) to send a shock through the powertrain, upsetting the car's balance. This causes the rear wheels to slip. The foot should be at an angle so the brake and gas may be pressed as well, this being needed to control speed and stop from spinning out in the drift.

Weight transition techniques
These techniques employ a further concept of weight transition. When a vehicle has the load towards the front, the "lighter" back is easier to steer, causing an oversteer condition that can initiate a drift.
Braking Drift - This drift is performed by braking into a corner, so that the car can transfer weight to the front. This is immediately followed by throttle in a RWD car causes the rear wheels to lose traction. FWD cars can also use this technique as it does not depend on the rear wheels being driven.
Inertia (Feint) drift, or Scandinavian flick - This is done by transferring the weight of car towards the outside of a turn by first turning away from the turn and then quickly turning back using the inertia of the rear of the car to swing into the desired drifting line. Sometimes the hand-brake will be applied while transferring the weight of the car towards the outside to lock the rear wheels and help the rear swing outwards. This type of drifting causes the car to accelerate faster afterwards, because of momentum built up while drifting.
Kansei, Lift off, or Taking In - By letting off the accelerator while cornering at very high speeds, cars with relatively neutral handling will begin to slide, simply from the weight transfer resulting from engine braking. The drift is controlled afterwards by steering inputs from the driver and light pedal work, similar to the Braking drift.

Other techniques
Dirt drop - This is done by dropping the rear tires off the sealed road onto dirt, or whatever low-grip surface borders the road, to maintain or gain drift angle. Also colloquially called "Dirt Turbo".
Choku-Dori - This is done by swaying the car's weight back and forth on straightaways, using countersteer and throttle to maintain a large angle. This is a show maneuver that usually involves many cars following the same line.
Ski Drift - This extremely difficult and dangerous drift is done by drifting while the car is on two wheels. If not done perfectly, the car will likely roll over and crash.