Trade-offs
CRTs - For CRTs the main tradeoff is size. You'll notice that CRT-projection TVs are huge--period. However, if you have the room and don't mind the behemoth size, they are definitely the cheapest choice per inch, the technology has been around for a long time, and the viewing angle is great. Burn-in is possible but unlikely on these sets.
Burn-in
Burn-in is when an image is left on the TV for a long time and is actually "burned" into the screen. For example, if you paused a video game on the TV and left the TV for a considerable amount of time, the image from the video game may still faintly appear on the screen even after the video game is turned off.
DLP and LCD Rear Projection
Some people suffer from the "rainbow effect" with these televisions. The rainbow effect is when you can see what looks like a "rainbow" on the TV either when looking at it directly or as you look away from the screen. If you're one of the few that can see it you'll notice it. However, the 3-LCD TVs and the 3 chip DLPs mentioned above eliminate the rainbow effect. Another tradeoff with these TVs is that they have a lamp inside which generates the light for the image. Just like ordinary bulbs, these lamps will burn out eventually and have to be replaced. Unlike ordinary bulbs they cost a lot of money to replace, anywhere from $200 to $400 a pop. Also, with moving parts like color wheels there is a higher chance that things can break over time. This doesn't mean they will break down sooner than an LCD or Plasma, you could easily have 6+ years of viewing with no problems and the technology has already gone through numerous generations. A benefit of these TVs is that the picture quality is great, they are not affected by Burn-in, and they are a lot smaller than CRT projection TVs and are cheaper than LCD and Plasma. Screen sizes are also large.
LCD
The main benefit of LCD TVs is the size. Slim and lightweight compared to other options. They also do not suffer from the burn-in problem. The tradeoff with LCD is per inch they are the most expensive choice (however, the price is coming down by the day and is catching up to plasma) and some have a slow refresh rate which can result in something called ghosting. What is Ghosting?: To display a moving image the TV redraws the screen many times over a short period of time. If the TV cannot refresh the screen fast enough you may faintly see the previous image for a split second. Even for the short period of time you will notice the previous image(s). It is more noticeable when displaying images that change at a fast rate such as during fast action scenes or during video game play. I wouldn't worry too much about this nowadays as newer sets are coming out with faster refresh rates. (Most manufacturers refer to this as the Response Time. A lower response time is better. eg 8ms response time is better than a 12ms response time.)
Plasmas
Just like LCD TVs one of the main benefits of plasmas are their size. Slim and lightweight. Unlike LCD TVs, Plasmas do suffer from the burn-in problem (which depending on who you talk to is an over exaggerated problem). Per inch plasma TVs are cheaper than LCD but more expensive than CRT projection, DLP TVs, and LCD Rear Projection. Plasmas do not have the ghosting problems, as the refresh rate of Plasmas is comparable to that of CRT televisions. As you can see, Plasmas and LCDs look similar but technically are almost opposites of each other.
Closing Words
I think the best bet when looking to purchase any of these TVs is to first consider your situation. How much money are you willing to spend, what will you mostly use the TV for, and how much living room space can you spare? Secondly, actually go out and take a look at each TV and decide which one looks the best to you. Try to visit different stores to look at the same TV (identical models) to confirm that the TV always outperforms from your perspective. Finally consider the tradeoffs mentioned above and decide what makes the most sense to you. As an example, you don't have much room to spare but are looking for a large screen TV and are willing to pay top dollar for it. Looking at the tradeoffs you decide that CRTs are too big. You definitely want to play video games but are afraid of the burn-in problems of Plasma. You're not sure if the refresh rate of an LCD will be fast enough and plus they don't have the size you're looking for. So you decide on DLP, LCD Rear projection or Plasma. All 3 offer the larger screen size but you take a visit to a few different stores and feel that the LCD rear projection TV looks best to you.
Finally, you may also want to take a look at the differences between EDTV and HDTV.