A wider tire will give your car better grip on dry pavement, but your gas mileage will suffer. Too wide and your tires may rub your car's body, destroying the tire. Narrow tires are better for traction in winter weather. But go too narrow and you're creating a safety risk. Your car relies on how much rubber touches the road to keep you going in the right direction. Unless you're trying to do some serious experimenting or you are being advised by a pro, you should always go with the size tire the auto manufacturer suggests.
Aspect Ratio You learned back in the Tire Ratings intro that a tire's aspect ratio determines how tall the sidewall is. A lower, stiffer sidewall will greatly improve cornering, but will make your car's ride a little harsher. That's why a Cadillac has a nice big bubble tire while a Porsche looks like somebody put a black rubber band around the rim. Changing the aspect ratio too much can throw your speedometer off, which is why some people go a a Plus-One setup. This way you have the benefit of solid cornering and you'll know how fast you're going.
This is pretty much non-negotiable. If your car has 16-inch wheels, you'll need 16-inch tires. The only way to change this is to buy new wheels, for instance if you were going with a Plus-One or even a Plus-Two setup.