Your vehicle's tires are the only thing that touches the road. No matter how great your car is, bad tires will ruin your car. It is all about traction. The more worn your tires get, the less traction they can develop. You wont be able to safely stop your car in the same distance. Turning and steering response will be diminished. Even the ride quality can go down, making it less fun to drive your car. Ask an racer what the most important part of their car is and they'll say the tires.
While mileage is a common determining factor in your tire's life, age is just as big. The rubber will eventually break down leading to decreased performance. All tires have a DOT date code printed on the sidewall. It will typically be a small oval shape with four numbers. The first two numbers are the week of the year and the second two are the year. For example: you may see a date code that says 4215. This means this particular tire was produced in the 42nd week of 2015.
Generally, once a tire is over 6 years old, its a good idea to plan on it's replacement. Regardless of miles. Also, proper maintenance can help extend tire life. Checking tire pressure regularly will prevent them from going low and wearing sooner. Having your vehicle's alignment checked and corrected every few years will ensure the tire is contacting the ground properly.
Tires are also one of the greatest performance modifications you can make. Switching from all-season to summer performance tires can yield great results. Check out this video for more info: https://www.youtube.com/watch?v=YhwD-b0d9DY