The ideal approach is to use four tires that match (same make and model). If you can't do that, then it's essential to have four tires whose performance characteristics are as close to each other as possible, so that the handling is predictable.
Let's look at an example and you can see how mismatched tires are a problem. Let's say you have different tires on the front vs rear of an NSX. On the front, you have the Falken Azenis RT-615, which is incredibly sticky on dry pavement, but is mediocre in rain. On the rear, you have the Kumho SPT, a budget tire that is just okay on dry pavement, but is excellent in rain. As you're driving down the road on a nice day, the front tires will grip better than the rear. As a result, the car has a tendency to "oversteer", meaning that, if you take a turn too quickly or too suddenly, the rear end has a tendency to swing around on you, causing the car to spin. What's worse is that the handling can then change if it starts to rain; then, the rear end grips better than the front, and when you turn the steering wheel to try to take a corner, the car has a tendency to "plow" and go straight ahead instead of going where you turn it. Unpredictable handling can be even worse - in fact, much worse - if you mix tires on the same axle, which I don't recommend at all.
In order to prevent this from happening, you need to have tires that all match, or are as close in performance as possible. That does NOT mean that the brand needs to match! If you have, say, the Goodyear F1 GS-D3, a top-of-the-line summer tire, on one end of the car, your best bet would be to use the same tire on the other end, but if not, you would be much better off with a similar tire, such as the Bridgestone Potenza RE050A Pole Position, that brand's top summer tire, than with a dissimilar tire, such as a Goodyear RS-A, a moderately-priced all-season tire.
If you have high-performance tires on some of your wheels, don't put cheap, low-performance tires on other wheels on the same car.
