Yep, I can help you here, because it can certainly be a mess of terms and jargon tossed around, and sometimes it feels like those tossing them around the most do it so that they can feel more intelligent than the rest of us! LOL
Here are the basics (and when I'm done, you'll know more of why I personally recommended a plasma set to you!)
1. Regarding LED and LCD tvs - LCD tvs usually have a fluorescent backlight. The newer "LED" or "LCD-LED" sets are still LCD panels, but instead of fluorescent backlights, they use LED backlights. Primarily, these are done for performance reasons - higher efficiency by up to 40% - less heat, darker blacks, a little better viewing angles, thinner, lighter. The business reason they did it was to sell higher-margin TVs.
2. Regarding edge-lit and full-array backlit LEDs - Again, this is also all about the backlight. Full-array backlighting has the best picture quality of the two, because - as the name implies - there is a full array of LED backlights on the back of the LCD panel, shining through it to the screen you're watching. It results in a more evenly-lit panel from corner to corner, with no bright spots anywhere. The downside? It's very expensive to install all those hundreds of LED backlights in the TV, as you can imagine. The other choice is edge-lit. As the name implies, the TV is lit by LED light sources positioned in a ring around the edge of the LCD panel, firing in towards the center. The downside is that it leaves a bright halo effect that shows up most often on dark or dim scenes. The upside? It's a lot cheaper to implement than full-array backlighting.
3. About Local Dimming - Local dimming happens in software, and all it means is that the set can control its LED backlights in zones, to give it more fine control over the black level. Here's a practical example: let's say you're watching Star Trek, and you're watching an inky-black star field. On an LED TV with Local Dimming, the TV will turn off power to the backlights in those zones that are in those pitch black areas of the screen, which makes them even darker! The effect is amazing, especially on a really good Blu-ray. The downside to it is the cost to implement. Very few TVs feature it because it's very expensive to implement. Sonys HX929 did it well at about $4000 for a 55" set. Sharp's Elite sets do it for $4500 for the 60" and $7000 for the 70"!!
So, all these steps are good and all, but by and large, they are there to catch up with where good plasma sets have been since 2008, as far as black levels and viewing angles go. Quality plasma sets represent the best bang for the buck in this space.
I hope that helps!