In order to maintain optimum control and stability for your vehicle, you should use identical tires with the same tread pattern, size, and construction. Generally speaking, you should not use a mixed or mismatched set of tires on your vehicle unless the tire and/or vehicle manufacturer specifies that it is acceptable to do so.
Examples of mismatched tires include mixing winter and all-season or summer tires, tires with different tread patterns, tires from different manufacturers, run-flat tires with non-run-flat tires, and tires with different construction characteristics or sizes. The only exception to this would be if your vehicle has a staggered fitment with different sized tires on the front and rear axles of the vehicle.
In addition to maintaining optimum control and stability, having a uniform set of tires allows you to regularly rotate them. This will enable you to have even tire wear, thus extending their life and maximizing your investment in your tires.
Although it is ideal to have your tires wear evenly throughout their lives, it is not always possible, even with regular rotations. Tires can wear unevenly if your vehicle has staggered fitments on the front and back, if your vehicle has an alignment or other mechanical problem, if your vehicle is insufficiently maintained, or due to driving conditions. You may also lose an individual tire due to puncture or other irreparable damage. In either case, you may be forced to consider purchasing one or more tires to replace the worn and/or damaged tire(s), or having to buy a complete set and forfeiting the remaining life on the non-damaged tires.
Obviously, it is best to replace all the tires at the same time because it will restore the handling and traction to optimum levels. However, due to financial reasons, this is not always possible. On the other hand, the less-expensive alternative of replacing two of the tires requires choosing exact, equivalent, or alternative tires.
If you decide to replace just two of the tires, the best choice is to replace them with the exact same brand, style, and size as the tires already on the vehicle. This is wise because the tire's physical dimensions, internal construction, materials, and tread design will be nearly identical to the tires you are replacing. If it is not possible to purchase the exact same tires as those currently needing to be replaced (for example, the tire is no longer available), then the next best option is to choose tires in the same performance category with equivalent load rating, speed rating, handling, and traction characteristics as the original tires. The final option of choosing alternative tires that do not match any of the characteristics of your original tires should only be used as a last resort or in an emergency situation. This is because the use of tires with different performance characteristics, load ratings, speed ratings, or sizes can cause the vehicle's handling to become unbalanced, especially in adverse weather conditions.
In summary, always try to maintain an even set of tires that are identical in brand, model or style, size, tread pattern, and tread depth. Anything else will compromise your vehicle's comfort, handling, and safety characteristics.
For more information on how and when to properly rotate your tires for even wearing, see the articles entitled Rotating Your Tires and Tire Rotation Patterns.