Clinchers are tires that are open on the side facing the rim. They use a separate tube to hold air, and they have a bead that holds the tire under a "lip" on the rim.

Tubular tires are round and closed all the way around the integrated tube. They are made of textile with tread material on the outward-facing side. On the inside, they are sewn closed (which is why tubulars are sometimes called "sew-ups") and glued to the rim, which has a closed concave outer surface.

Because the glue takes some time to cure, if you have a flat, you pretty much need to have a spare wheel ready, which is why hardly anyone uses tubies outside of racing. The advantage is that they're as close to perfectly round as possible - which gives them great handling characteristics - and they can take a much higher pressure than clinchers.