Nowadays, trying to keep track of what day it is can be a challenge, let alone remembering the last time you rotated your tires. Often, infrequent tire rotation (or an improper alignment or air pressure) can lead to uneven wear between the front and rear tires, causing many drivers to replace two tires instead of four. In these situations, it can be confusing as to what tire to buy and where to have the new tires installed on the vehicle.
In all cases, it's best to have a set of four tires that are the exact size, brand and model. While matching the tire size and performance category is a step in the right direction, without an exact match of rubber compound, speed rating and tread design you can be risking unpredictable handling and braking performance, especially in wet conditions. Given that different tread designs will shed water and snow with varying success, a mix of two different tread patterns between the front and back axles is something that is not recommended. When the exact matching tire is not available (due to the model being discontinued or out of stock), it is always helpful to seek the advice of a tire professional to find the closest match possible. In some cases, the safest option may be to replace all four tires instead of mismatching models that would not be "close enough." While the natural desire is to choose the best tire based on reviews and testing, without a matching set of four, the best reviewed tire is not necessarily the best match to what you have already. For a better understanding on replacing less than four tires, read "Mixing Tires."
Regardless of where the worn tires were removed from, when replacing a pair of tires, the new pair should always be installed on the rear axle. Installing the new tires (with full tread depth) on the rear axle will ensure that any differences in traction between the two pairs of tires will not lead to a dangerous oversteer situation. In an oversteer situation, the rear tires will begin to slide and can be very difficult to correct. However, if the front tires have significantly less tread depth than the rear tires, the front tires will begin to hydroplane and lose traction on wet roads before the rear tires. While this will cause the vehicle to understeer, you can correct this by releasing the gas pedal, staying off the brake and allowing the vehicle's weight to shift forward and load on the front tires to improve grip while you regain control.
For more information on properly installing a new pair of tires, read "Where to Install New Pairs of Tires?"