Yes and no. If you want to rotate the tires on your vehicle to maximize tread life, then all the tires have to be the same size. Rotating the tires every 5,000 to 7,500 miles is a good idea, especially on front-wheel drive cars and minivans where the front tires tend to wear out long before the ones on the back. Wide, low profile tires also tend to develop unusual wear patterns if left in the same wheel position for their entire life.
If you don't plan to rotate your tires, however, then the front fires can be a different size than the ones on the back. But the tires on both front or both rear wheels must be the same size for proper handling and braking. Mismatched tires side-to-side can cause a vehicle to lead to one side and/or to pull when braking.
CAUTION: Never mismatch types of tires either on the front or rear wheels. A bias ply tire should never be paired with a radial tire, and vice versa. Likewise, tread design, belt type and overall tread wear should be the same (or similar) side-to-side for proper handling, steering and braking. Intermixing different brands and styles of tires may cause similar problems in some instances.
As for the size of replacement tires, some people will try to use anything that fits. This may not be a good idea because a vehicle's handling, steering and braking characteristics can all be adversely affected by using tires that are too large or too small for the application.
Some people may want oversized tires on their rear drive wheels to improve fuel economy. Switching to a larger diameter tire reduces the number of revolutions per mile. But it also affects the accuracy of your speedometer and odometer readings. Others may want wider or larger tires on the rear drive wheels to improve traction or to achieve a special kind of "look." Wide tires, however, typically provide reduced traction on wet roads.
Clearance problems can arise, too, if the tires are too wide or too large. They may rub against the body when turning or when the vehicle hits a bump. This can not only damage the vehicle but also the tires, possibly causing a blowout.
On cars and trucks that have antilock brakes (ABS) equipped cars and trucks, most vehicle manufacturers say replacement tires should be the same size as the originals. This is necessary because the diameter of the tire affects how the wheel speed sensors read, which in turn affect the operation of the ABS (and traction control) system. Changing to a larger or smaller diameter tire, or installing different sized tires front and rear can upset the operation of the ABS system. This may create braking problems and/or cause the ABS warning light to come on (which means the system is deactivated).
Vehicles with all-wheel drive or full-time four-wheel drive must also run the same sized tires front and rear to maintain the proper drive relationship between axles. If one set of tires is larger or smaller than the others, it will create slippage between the front and rear axles that will accelerate tire wear and adversely affect handling.