WELDED DIFFERENTIAL
Welding your factory diff is hands down the most affordable option however it is also the hardest on your drivetrain. When you weld your factory open differential you are essentially locking the wheels to make them rotate at the same speed all of the time even while coasting. This will allow your car to drift and handle similar to having a 2-way differential however under tight cornering at low speeds like in a parking lot for instance you can experience very violent shaking and wheel hop/chirping from your rear tires. This occurs because the tire on the inside of the corner rotates at a slower revolution than that of the outside tire. An open differential allows the inside tire to rotate at a different speed without any ill affects however once it has been welded the inside tire is forced to rotate causing it to skip across the pavement causing excessive noise from the rear of your car.
1.5 WAY DIFFERENTIAL
1.5-way differentials lock the wheels 50% during deceleration and 100% during acceleration. This helps compensate wheel chatter and drivetrain strain during turning by allowing clutch or viscous discs in the unit to slip under no or light acceleration however also lock up the rear tires under hard acceleration. A 1.5-way differential is generally a good choice for a car that will see a lot of daily driving or a novice drifter given the fact that is is easier to regain control of the vehicle once you let off the accelerator.
More info and pricing on 1.5-way differentials can be found here.
2-WAY DIFFERENTIAL
2-way differentials lock the wheels 100% both under deceleration and acceleration while still allowing discs to slip in the unit to help combat wheel chatter and excessive stress under slow cornering. 2-way differentials are generally the norm when it comes to competitive Formula Drift or Pro-Am cars given the ability to maintain drift at slower speed or off throttle. Being as the rear wheels lock even under deceleration 2-2ay differentials are not recommended for cars that will see a lot of daily driving use or inexperienced drivers. For instance a downshift in the rain with an inexperienced driver can cause the rear end to lock up throwing the vehicle into an uncontrolled slide.
More info and pricing on 2-way differentials can be found here.