"'First off, it isn't really HD unless it is 720p or more. At the moment there are only 3 HD phones: the Galaxy Nexus, the Rezound, and the LG Spectrum. The nexus is OLED and the other two are LCD. (The Razr is OLED too, but not HD resolution)
There are two main standards; LCD and OLED.
LCD is the conventional standard. This is like what you've seen in laptops. The pixels are all lit with a lightsource from behind. The pixels themselves block the light...so a pixel that is "off" will show completely white. All you will see is the light part if it. Because of this, you can never get true blacks. If you turn off all the lights in a room, a "black" screen will still glow. LCDs have the advantage of better color reproduction on average compared to OLEDs, just because the tech is more mature.
OLED stands for "organic LED". The display has no backlight. Each pixel contains biolunenescent bacteria which glow when activated. So when you want black, you just turn them off. Because of this, OLED displays always have deep and perfect blacks. In a dark room, a black OLED display cannot be seen at all. They also tend to be the brightest displays right now and in theory are more energy efficient than LCD. Their main weaknesses right now are ghosting (a trail left behind moving pixels) and burn in (where an image on the screen a long time leaves a shadow of itself after you move it). They also tend to have color banding in gradients and some people consider the colors over-saturated (probably just caused from the contrast with the deep blacks).
Overall there is no "best" between the standards. Both have merits. When OLED technology fully matures, it will probably surpass LCD, and all the weaknesses listed above will not be there anymore. But it hasnt reached that point yet.
