On average, an edge-lit LED TV is the thinnest, with a width of only 1.8". Plasmas are not so bad anymore, being an average of about 2.5" deep.
Thickness is mostly for appearance and impressing your friends. A very slim television looks a lot better in your living room than a bulky one does. It is also easier to mount on the wall, because there will be less bulk in the back.
You should not base your buying decision this factor alone, unless the aesthetic of your television and how it fits into your room are very important for you. Consider thinness a bonus if you are buying an LED TV.
A plasma television is heavy. Each cell of gas representing a color of a pixel has its own chamber, nestled between two panels of glass. The glass must be strong to stand up to the difference of pressure between the ambient air and the gas chamber. An LCD screen is lighter because it does not need panels of glass. It can be simple plastic.
Like the thickness, the weight of the television should not be a main factor when choosing your television. However, a lighter TV means an easier time moving it around or hanging it on the wall. The lighter it is, the less likely you are to break it.