Some people avoid plasma because of worry about screen burn-in. Screensavers and other improvements have made burn-in less of a worry, but the phosphor in the screen can still wear out over time. Still, a modern plasma screen should last you for many years.
Plasma TVs start at 32 inches and go up from there. Plasmas with 1080p resolution start at 42 inches. Buying a plasma unit is generally a straightforward decision. Once you've decided you want plasma, you're free to choose your new TV based on size and price.
OLED, or organic light-emitting diode, displays have finally hit the scene and look set to outperform both LCD and plasma in image quality and electrical efficiency. Unfortunately, for the moment there are very few sets available and they're expensive. An 84-inch display can cost $25,000 and even 55-inch units will probably be more than $10,000. Unless you're independently wealthy and have to have the latest technology, I'd give OLED a pass for the next few years.