Trail Tires
The most typical mountain bike tire, these are designed for grabbing traction on a variety of surfaces, including dirt, mud, grass and loose rocks. Trail tires may consist of nothing but a series of large knobs, or they may have a more refined pattern, but the purpose is the same. For high performance on very rough terrain, serious mountain bikers reduce their tire pressure so that the tires will form to the surfaces they cycle on. Specialty tires for particular surfaces, such as mud or rock, are available at cycle stores, but these are generally only necessary for competitive mountain bikers.
City Tires
While a mountain bike is designed for sturdy trail riding, many cyclists also use them in the city. The sturdy frame is excellent for hard riding in the city, but trail tires aren't necessary on pavement, and the knobs and deep grooves for trail riding add unnecessary weight to the tires. Many cyclists maintain a separate set of tires for the city and for trail riding. City tires are much smoother than trail tires, offering little in the way of teeth or knobs. Some city tires are completely smooth, offering the least rolling resistance. Most, however, over some minimal grooves for maintaining traction on wet pavement or gravel.
Hybrid Tires
These tires are the best for casual mountain bikers who may transition between riding on trails and in the city, but who do not want to maintain two separate sets of tires. These tires typically have knobs and grooves on the sides, for trail traction, but have a center section that is relatively smooth, and designed to roll well on concrete surfaces, such as roads and sidewalks. The extra traction will also kick in when you make hard turns, which can be very helpful.
Materials
Just like automobile tires, most bicycle tires are rubber, reinforced with steel fibers. This makes them sturdy, long-lasting and harder to puncture. It also carries a weight premium with it. Newer, lighter tires are available woven with Kevlar threads, which can save several ounces per tire. This is a substantial difference for some cyclists, but Kevlar tires have significantly higher price points.