It is of the utmost importance to always use the same size tires on a vehicle, unless the vehicle comes from the factory with a staggered tire fitment. Using the wrong size tires on a vehicle can cause any number of problems, from minor inconvenience to major mechanical damage.
If nothing else, installing the wrong size tires on your vehicle can cause handling issues. If the front end of your car has a different size tire on one side than the other, the vehicle can pull to one side, creating a dangerous situation should you remove your hand from the steering wheel. Tires can also wear faster if one side has a larger tire than the other.
A more dangerous situation can arise from installing different size tires on your vehicle; contact with suspension parts or body panels. Installing different sized tires on a car may cause irregular suspension travel and may result in the tires striking the suspension parts or the wheel well. Besides the obvious damage to suspension parts or body panels that this may cause, it can also cause damage to the tires themselves resulting in faster wear. In a worst-case scenario, the tire may be cut and blow out while traveling down the road.
The biggest drawback to using different size tires on your car is the potential for significant transmission damage. This typically occurs on all-wheel drive vehicles or vehicles where different size tires are used on the drive axle. Using different size tires on the drive axle causes the transmission to continually be in a bind, since larger tires rotate slower than smaller ones. Ideally, the wheels should rotate at the same speed unless the vehicle is turning. Over time, using different size tires on the vehicle will cause the transmission gears to strip, resulting in expensive repairs. It's far easier to just use the proper size tires to begin with.
As long as the vehicle isn't an all-wheel drive or four-wheel drive vehicle, it's usually okay to use different tire sizes between the front and back, provided the larger tires don't strike the suspension or any body panels. Be aware that using different size tires between the front and rear may affect the handling characteristics of the vehicle as well as require recalibration of the speedometer. In short, the short-term savings of using a different sized tire is not worth the substantial cost that may be incurred from repairing any problems that may arise in the future.