Actually, all these technologies are "old"
DLP- Neat reflective technology developed by Texas Instruments in the 70's finally was able to be produced lately. Uses a chip with really small mirrors on it. The light from a mercury lamp is either split by a prizm, or in the cheaper models, passed though a color wheel, and via reflection off said chip or chips, displayed on the screen. This is a projection type type technology and comes in a rear display or projector form. Best picture of all the rear view projectors, but you can hear the color wheel and cooling fans.
LCD-
Rear projection screen units pass the light from a mercury lamp(split by a prizm in most cases) though the LCD panel(s) and on to the front screen . Projectors project to screen.
Direct view LCD units, like your computer screen are relatively new because the yields have improved drastically in making these screens. These are the lowest power consuming devices of all mentioned, and IMO the best looking and have an awesome cost for feature ratio!
Plasma-Direct view only. Color was recently added to this 70's tech due to technology. The units are heavy, difficult and expensive to work on, and consume tons of power. This is also the only technology that is listed here that actually uses lead and phosporus in the display panel. Plasma works by having each pixel being 3 gas discharge cells. Each cell needs to be prepped and either discharged, or un-prepped. This takes a huge amount of power, and generates a fair amount of heat. Because the backbone of plasma color technology is phosphorus, it suffers from the same issues as a CRT as far as aging of the display over time and displaying of still pictures.
From my dime, if I had to go by a TV, it would 100% be a LCD set. Samsung makes a nice set, and parts are readily available. Sharp also makes a nice LCD set, but parts are only available to dealers for the Aquios line. When the place I worked at sold Vizio, the only way to get it fixed was though Vizio. Depending on the size of set, a number of the computer companies, like HP have gotten into the game and stand behind their products rather well.