Overall advice: buy a 1080p set. Some people will say buying 720p/1080i is a good way to save a little money, but the truth is that 1080p sets are the way to go, especially if you’re EVER looking to hook a PC up to the TV.
Plasmas have the best picture quality you can currently buy. They’re vibrant, beautiful, and don’t suffer from limited viewing angles. On the flipside, they do require a little bit of care to minimize image retention. They also draw more power than LCDs and are substantially heavier. Plasmas are also the absolute best DEALS on the market, since they’re cheaper than LCDs while still offering a better picture. Panasonic is the brand to look at. I can’t really recommend specific models, but even the cheapest 1080p sets in their range should be good.
LCDs are lighter, thinner, and a little more idiot-proof (since image retention isn’t possible). The image quality can be darn near as good as plasma (some would even argue it’s better on some sets), but you’ll pay a lot more for the good lookers. Many LCDs have “120hz” picture, which is a nice feature because it allows you to watch both 60fps TV content and 24fps movies at their native framerate. Such sets also usually have a fancy motion mode where it blurs between frames to make motion look smoother. Some people love the effect, some hate it. High-end LCDs also have more bells and whistles like the ability to stream internet content or networked photos and such automatically. In LCDs, I’d say that Sony and Samsung are really the two brands to consider.
If you’re a movie buff, I’d say that a Panasonic plasma is the obvious choice. You’re not going to find anything better for movies. For TV shows, it’s a wash. If you like videogames or want to hook a PC up to the screen, LCD might make more sense. Also, if you’re looking to wallmount, very large plasmas can get VERY heavy and require some serious hardware.