Unlike LCDs, most people had never heard of plasma screen displays before HDTV came to the public eye. Some of the first HDTVs were plasmas, but those early impressions were not the best. The first plasma televisions were expensive, fragile, and prone to burn-in, a phenomena where an image displayed for too long a time becomes permanently etched into the television. Plasma displays are also a fundamentally less understood technology. A plasma display consists of numerous tiny cells filled with a mixture of inherit gasses which, when charged with a bit of electricity, become excited and emit light. These cells make plasmas a bit heavier and thicker than LCD displays, and unsuitable for the common household usage that makes the LCD so familiar.
That said, modern plasma displays have a number of advantages over LCDs. The largest advantage is in black levels. As explained above, LCD displays have poor black levels which causes the detail in dark images to be lost. Plasma televisions do not suffer from this problem because there is no back-light. Each individual cell produces its own light when activated - and this means that each individual cell can be turned off completely, creating a deep level of black. Color production also tends to be above-par on plasma televisions, with even the basic models displaying a high degree of color accuracy. These features make plasmas the display of choice for most videophiles. And while plasmas were once expensive, they are now more affordable than LCDs. Pioneer and Samsung both sell 42" plasma televisions with an outstanding picture for under $1000 bucks.
Plasmas tend to be less practical than LCDs, however, in that they have glossier displays and are inherently less bright. This makes them more suitable in a dedicated home theater than a den or family room. They are also heavier and bulkier, which can be a problem in various situations. Screen burn-in is no longer a major issue, but plasmas are still unsuitable for anyone who might be displaying a static image for days at a time. Finally, plasma displays do tend to have lower resolutions than LCDs - for example, most 42" plasmas are 720p, while most 42" LCDs are 1080p.