What Matters to Me?
Obviously, the size is the single most important piece of info on the tire - get it wrong and it might not even fit. Size includes the width, aspect ratio and wheel diameter. These three measurements will be universal no matter what brand of tire you're looking at. There's a lot of room for experimentation and error in this realm, so tread with care. Be sure you know what happens when you change sizes before you deviate from the standard.
The average driver doesn't need to be too concerned with a tire's speed rating. There aren't any tires out there that will blow at 70 mph. However, there are tires that will blow at 120 mph, so if you are planning on driving at those speeds, you better pay attention to this number.
Loads and Pressures
These ratings aren't going to be too important to you, either, but there are a few interesting points of information here. For instance, if you are comparing two tires that are the exact same size, but one of them has a higher maximum tire pressure, this tire may get slightly better gas mileage since a firmer tire offers less resistance. Your load ratings will only be important for trucks. If you are hauling some serious tonnage or pull a heavy trailer you might want to invest in a tire with higher load capacity. This will keep your rear tires from pancaking and improve stability.
Treadwear and Traction Finally, some numbers that are useful again! A higher treadwear rating means you'll be able to drive more miles before you need new tires again. And tires with higher treadwear ratings tend to offer gas mileage advantages because they are made with slightly harder rubber. which means less resistance. The downside to harder rubber is less cornering performance, so if it's important to you to be able to carve up some mountain roads don't go for the high treadwear rating, go with the higher dry traction rating instead.