Bicycle tires, particularly tires for road or racing bicycles, come in two different types. Clincher tires are open on one end to fit a tube and have a bead that fits into the edge of the rim of your wheel. Tubular, or sew-up, tires are made of layered materials and are a one-piece tire and tube combination that is usually glued to the surface of a rim designed specifically to hold a tubular tire. Each of these has benefits and drawbacks, basically boiling down to cost, performance and ease of maintenance.
Tubular tires are designed to hold a tremendous amount of air pressure -- up to 140 pounds per square inch. This makes the tires firmer when inflated, making them roll with less friction over a wide variety of surfaces. Clincher tires are not generally designed to hold the same volume of air pressure that tubular tires can. However, higher quality clincher tires have been developed in recent years that allow these tires to hold up to about 120 psi, making the functional difference almost nil.
Because tubular tires do not have a metal bead embedded in the edges of them like clincher tires do, they tend to be significantly lighter and, as a result, preferred by racers who fight tooth and nail for every spare ounce. Recent advances in clincher design utilizes Kevlar instead of steel in the tire beads, making this difference much less of an issue for all but the highest level riders.
This is where the clincher tire takes a clear lead in the debate. To repair a clincher flat, all you need to do is remove the burst, pinched or punctured tube, replace, remount and inflate it, and you are riding again. Since tubular tires are glued in place on your rims, they take much more work and time to repair. The tire needs to be mounted and glued, and then you need to wait for the glue to dry before you inflate the tire and start riding. You can replace a tubular tire without gluing it, but you need to avoid cornering at high speed until you've glued your new tire.
High-level bike racers prefer sew-up or tubular tires for a variety of reasons, including lighter weight, reduced friction and added precision in cornering at high speeds. But most racers at this level have the advantage of a support team following in a vehicle filled with replacement wheels ready to go. Recreational riders who need to repair their own punctures on the fly tend to prefer clincher tires.
Tubular tires have a much higher price range than the average clincher tire. Tubulars can go for $25 at the low end to more than $100 at the high end. On top of this, they require specifically designed rims that tend to be more expensive because of the target audience of competitive cyclists. Clincher tires designed for training can run between $15 and $50 and racing clinchers can be found for $20 to $60 or so.