Flatscreen is kind of a tricky word, because it can be used to refer to a flat-panel TV (like an HDTV), or a tube TV with a flat screen as opposed to the old fashioned curved screen. Just stick with calling it HDTV.
Now then, plasma and LCD are the two main types of HDTVs, and its debated which one is better.This site explains some of the advantages of disadvantages of each, although the differences are becoming less and less noticeable as each technology improves.
If you want my recommendation, for a good price/performance ratio, go with Vizio. IMO, their TVs look awesome, and they are generally priced quite a bit lower than similar TVs of other brands.