A regular tire is just what the name is, your standard high-grade rubber road-worthy tread and grips. For a driver who is asking their car to get from Point A to Point B on normal paved roads in temperate weather, you'll probably be just fine with regular tires.
Snow tires are made with a softer rubber that allows them to more firmly grip snow and ice. They're also produced with an elevated number of snipes in the threads for even more traction. There are also studded snow tires which are suggested for smaller vehicles, as they greatly increase road friction (and, in turn, safety).
Also called mud tires, these tires are engineered with broad, thick treads to allow for breaking through mud. There are sharp angles and patterns built into the tread to fling mud backward and away from the vehicle as it moves forward. Generally, these tires are also wider than a regular tire to prevent them from sinking, like a snow shoe works for walking atop snow.
High-performance tires are not just for sports cars. These tires are great for the driver who does a lot of highway driving. Like snow tires, they are made with softer rubber for greater traction on the road and to allow for sharper turns at higher speeds. However, the soft rubber means that the treads wear much faster than a regular tire. High-performance tires need to be replaced on a regular basis.
Run-flat tires are somewhat new to the tire market and you may not be familiar with them. Essentially, they are built to allow you drive further on a flat than you'd be able to do with a regular tire. Inside a run-flat tire is a firm inner cage made of plastic. A monitoring system will notify the driver when there's been a blowout, but the tire allows for the driver to keep driving and maintain solid control of the vehicle. These are great tires, but as they are new to the market, they're still pretty pricey.