LCD = Liquid Crystal Display - Liquid Crystals can essentially change their physical state and either block light or allow light to pass thru them. They make this change in response to an electrical signal applied to them. Liquid Crystals, themselves, were discovered in the early 1900s, but the first LCD wasn't pioneered until around 1968, the LCD allowed researchers at RCA to create extremely thin, small electro luminescent screens for watches and other electronics.
In essence, LCDs are "light engines" they work by selectively blocking or unblocking light from a Florescent lamp from reaching the viewer. Multiple layers of LCDs and color filters are then sandwhiched between glass panels to form an LCD matrix. By manipluating the matrix with an electrical field, it is possible to producing the on screen image Since the liquid crystals don't generate light themselves, (only block or allow it to pass) a florescent lamp, called a BACK LIGHT is placed BEHIND the screen to increase its brightness. This 'back light' is constantly shining, so it reduces the contrast of the LCD screen (the difference between the lightest and darkest parts of the screen) Even in a completely Black image, some light from the back light leaks thru.
Also, since LCDs must cycle between 2 PHYSICALLY states the rate at which the image cycles or "refreshes" is slower then in other TVs, causing what is called 'ghosting'
Plasma televisions, which were developed in the mid-late 90's are in some aspects are a simpler technology.
In physics, plasma is the 4th state of matter, and is the most energetic. Essentially a super heated gas which is so excited with energy that it glows (emits light), becomes polarized, and thus responds to magnetic fields. In labs, plasma is often moved around with magnets.
So Plasma TVs essentially manipulate small amounts of plasma to create an on screen image. Each pixel contains a small amount of plasma which is excited until it glows a certain color.
Since each pixels emits light on its own, there is no need for a back light to illuminate them.
So unlike LCDs, which do require a back light, Plasmas have a much higher contrast ratio and can produce much darker Blacks.
Also, since Plasmas do not have to switch between 2 physical states, like LCDs, they are also able to cycle onscreen images much faster then LCDs, increasing their "refresh rate" and reducing 'ghosting'
1st generation Plasmas were prone to "burn in" if left displaying the same image a long time (like pausing your DVD for an hour or two) Burn-in essentially is where the plasma would get locked at one color and you would always see a faint ghosted image of whatever was 'burned into the screen'
Subsquently refinements in plasmas design have nearly eliminated this issue, but many people still are leary of Plasmas for this reason.
The longevity of plasmas are also in question because they are essentially super heating gases until they glow, some plasma displays have burned out after 5 years.
When it comes to TVs - Due to manufacturing difficulties, LCDs usually top out around 37-42 inches. The majority of TVs above about 42 inches are usually Plasmas.
Price wise, LCDs are cheaper below 42 inches where as the opposite is true ABOVE 42 inches.
There are advocates of both technology.
Personally, I think it depends on how long you expect to use the display and what you care most about or watch the most.
If you plan on only having the TV 3-7 years, or watch a lot of action movies, and want a really large display (above 42inches) for the cheapest price -- go with Plasma
If you plan on keeping the TV 7+ years, have it running constantly, looking for 21 to to 37 inch TV and/or price is no object, then go with LCD.