This article (although now a little dated) explains the three TV's quite well.
You need to think of several things:
Power use: Plasma TVs are power whores, they will gobble up as much as they can suck through the power cable without it melting. LCD TVs, in contrast, sip away silently.
Contrast: Plasmas have better contrast ratios, but high end LED LCD TVs are catching up fast, so the difference is almost negligible (unless you're anally retentive). Regular LCD and ultra thin LED LCD TVs aren't quite as good - you will notice that black isn't really black.
Weight: Plasmas are again the elephant in the room - these beasts can weigh 60-110kg, depending on size. You will need help putting it in its permanent place. LCD Tvs are a lot lighter and can often be wall mounted and even my grandma could lug one around (although she is a bit of a cleptomaniac so maybe she's had practice?!?).
Latency: I.e. motion blur, especially noticeable when watching sports or "akshion" movies. This is where Plasma tellies used to rule the roost, but again, LCD TVs are catching up fast. You can now get LED LCD TVs with a 200Hz refresh rate. And considering that the human eye cannot process more than 60Hz, I think 200Hz is plenty. Especially when you consider that most visual media material is delivered at piss poor framerates (24FPS is nowhere near enough, Hollywood needs to stop being such cheap skates).
Basically what it breaks down to is money, ergonomics and aesthetics.
If you have little to spend, get the most expensive Plasma you can afford, whilst ensuring it has a good refresh rate. But if you fear you may have issues with weight, go for a regular LCD.
If you have tons to spend, then it is almost irrelevant whether you go for an LED LCD or a Plasma. The decision here is aesthetics. Weight is irrelevant too, as you can afford to get someone to lug it for you.
Oh, and when it comes to size, ignore the calls to get the biggest you can afford. The ideal size of TV is dependant on viewing distance - getting a massive 60" when your couch is only 2 meters away is counter productive as you
will see the pixels and wish you bought a smaller telly. Measure your seating distance from where you propose to set up your new TV and talk to the experts at a specialist store (and by this I do NOT mean a big box retailer).
Make a list:
How much have you got to spend.
How far away from the TV will you be sitting.
What features MUST it have (e.g. USB, how many HDMI inputs etc.).
Take this list and talk to a few different stores - you will quickly pick up on those who intend to sell you the best matched TV and those who have nothing but their comission in mind.