A clincher is a bike tire which requires a inner tube. It has two wires made of metal or kevlar which hook onto the rim.

A tubular is a tire which has a inner tube that is sewn into a tire. A Tubular is a tube while the clincer is 3/4 of a tube

The advantage of a tubular is that the tire plus tuube is going to be lighter, the ride is going to be nicer, the tubular wheels are going to be lighter becuase it lacks the need for extra material fo the tire to hook into. It is faster changing a tubublar tire, If you catch a flat the tire is less likely to roll of the rim... You can run higher tire pressures (not necessarily an advantage)

disadvantage... more expensive, more expensive. Each tire costs a lot more. If you catch a flat (unless you have a lot of free time) you have to get rid of the entire tire not just a tube. A litttle bit of a pain of the ass to mount the tire onto the wheel.

Clincher. - Cheaper, easier to find, more options, last longer
Disadvantage - heavier, not as smooth ride,

Of course modern clinchers are approaching tubulars in ride quality and there are quality tubulars which are cheaper. So to each his own.

One more thing that taku didn't explicitly point out. Tubular tires are mounted by gluing them to the rim. Mounting can be a real messy pain in the ass, especially when you're doing it for the first couple of times. It can be time consuming and really is a multi-step process. I find that I do things like plan to mount tubies days in advance. I make sure I stretch the tire for at least 24 hours, to the actual gluing and mounting, let the tire set for at least 24 hours..........then ride. Clinchers you can mount whenever, wherever, at a moment's notice. That's why I train with them.

BTW, changing a tubie in a race is so quick because you can take advantage of the fact that after ripping the flat of the rim, there's a ton of glue still stuck to the rim that will adequately hold the (pre-stretched) spare.

With tubies I get to use glue . and I just love the smell. And the spair tire behind the seat looks so kool. Tires don't cost that much and I, oh no train on them to gasp. My reflex rims dura-ace hub wheels are light ,I think and I would say bomb proof. Tufos tires with the sealant inside have done me fine. Some say clinchers are geting as good as tubies you are dreaming. I do live in an area with not much of a flat problem so that helps. You only live once. Go tubular once just for the fun of it.
Flame on all The Dirtball !!!