Prasiolite (or vermarine) is the name for any quartz crystal or cluster that is green in color. Although often cut as a gemstone, prasiolite is actually quite rare in nature.
Most prasiolites on the market are heat treated amethyst. These stones are sometimes called "Green Amethyst", possibly because amethyst is the starting variety, or possibly just because it sounds more valuable that way. When heat treated, the amethyst from most locations transforms into yellow or orange citrine. From a few locations, a green color results. These locations include Montezuma, Brazil, and Arizona. Naturally occurring prasiolite has been reported from Poland.
Prasiolite (sometimes spelled praziolite) can be confused with the similarly colored praseolite which results from the heat treatment of iolite, a variety of cordierite. Note the spellings: prasiolite is the quartz variety, while praseolite is the cordierite variety - not the other way around (which would be easier to remember).
Some colorless quartz crystals have sufficient inclusions of chlorite to result in a green color. However, this would not be considered prasiolite. In any event, the color is wrong - more of a dark grass-green than a leek-green. Note that green ghost crystals inside a clear quartz crystal is almost certainly a chlorite coating over an earlier quartz crystal, and not an example of quartz over prasiolite.
Prasiolite is only one of several quartz varieties. Other varieties that form macroscopic (large enough to see) crystals are as follows: