When you want to learn how to drive a rally car, you have two main options to choose from if you're living in the United States.
On the West Coast, there's Dirtfish. While on the East Coast you have the Team O'Neil Rally School. The latter's been cranking out a lot of videos lately, and each one provides tremendous insight and information into the world of rallying. In the latest clip, you're going to get a lesson in rally tires and how they differ from your road car tires.
Your street car needs to have a set of tires that are relatively quiet, provide a comfortable ride, and last as long as possible. So the tire manufacturers run steel belts and Kevlar bits on the contact portion of your tire, while the sidewalls are usually fairly soft. This provides a compliant, consistent driving experience.
Rally car tires have little support under the contact patch, yet the sidewalls are darn near bulletproof. You don't want a rally tire folding over on itself under hard cornering, or run a risk of removing itself from the bead and the wheel. Additionally, rally tires are far softer in terms of the rubber compound used. You don't need the tire to last as long and you want more grip when you're ripping down a gravel stage. The softer compound provides this but won't last as long as your street tire.
There's a lot to learn, and Team O'Neil Rally School does a great job of explaining it all.