Clincher tires are the most common type of tires used in cycling. They have an open design with a bead that hooks onto the rim. Tubeless tires, on the other hand, are a newer technology where the tire forms an airtight seal with the rim, eliminating the need for an inner tube. Tubeless tires offer advantages such as reduced rolling resistance and enhanced puncture protection. However, they require specific rims and sealant for proper installation.