I often speak to customers about mixing tires on their vehicle due to the fact that either their front or rear pair have worn out before the other pair. In most cases, they don't want the same type of tire.
There are a variety of reason why people don't want matching pairs, such as lack of availability of their current tires, price or they just don't like what they have been using. We always recommend matching tires, or if that isn't possible, to find a model that is the closest to what they have. Mixing different tire types is the equivalent of walking outside with two different types of shoes and running a marathon. You could do it, however you won't get the best results.
Take a look at the tread differences between the Bridgestone Potenza RE970AS Pole Position (Ultra High Performance All-Season) on the left and Bridgestone Potenza S-04 Pole Position (Max Performance Summer) on the right. You can probably guess the tire on the right is going to have better cornering, braking and non-winter traction. Mixing these two on the same vehicle won't give you the best and safest results.
If you do decide to mix tires, then match axle pairs is best when installing two new tires. The new pair should always go in the rear (assuming the vehicle doesn't use staggered sizes), regardless if the vehicle is FWD, RWD or AWD. Doing this assures you have the best traction possible in the rear. Having less rear traction than the front will increase the likelihood of oversteer, which is harder to control than understeer.