OLEDs are made from plastic, organic layers rather than crystalline layers like LCD. This means they are thinner, lighter, and more flexible. You could have an 80-inch high-def TV only a quarter of an inch thick. They can even be mounted on a plastic rather than glass base, and will be easier to produce in large sizes (movie screens? billboards?)
The are also brighter and do not need to be backlit like LCDS, because they make their own light rather than selectively blocking out light. This also means they have a much larger viewing angle than LCD (about 170 degrees). The lack of wasted light production also means they use less power than LCDs and produce truer black and whites.
They also refresh about 1000 times faster than LCD, so will reproduce motion better.
A pretty substantial package of benefits! So why hasn't OLED taken over already?
Right now, it is expensive to manufacture OLEDs. But competition in display technologies is so hot right now. Economic forces will no doubt make it attractive to mature this technology quickly, and it could end up cheaper to produce OLEDs thand LCD because crystals need not be grown and placed.
Another problem is that the lifetime of the blue organic components in OLEDs is currently pretty short (around 14,000 hours). Once again, I think the economic forces will overcome this relatively soon.
Currently, OLED is used in cellphones, digital cameras, PDAs like the Sony Clie, and a few prototype TVs. But as the video below shows, it will be used in pretty much any display application ( tvs, walls, clothes ) because developers are really throwing money and research hours at it. They believe it will be the technology to replace LCD: