Pilsner is a variety of lager. Other varieties include bock, dunkel, and Oktoberfest/Maerzen.
Lagers are made using bottom fermenting yeast at relatively cool temperatures over relatively long periods of time, while ales are made using top fermenting yeast at warmer temperatures over less time.
For a good and entertaining little history of pilsner, see http:
//beer.about.com/od/lager/a/PilsnerHistory.htm
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Lager beers use lager yeast and ferment slower at low temperatures. Most all of the large, industrial beers use this Germanic-style of brewing. Stout beers use longer roasted (dark) malts and ferment at warmer temperatures using ale yeast. Ales are more prominate in the English/Irish styles of brewing.
Draught beer isn't a particular type of beer. It's any beer that is served from the keg instead of in a bottle or can. Draft beer is exactly as it comes from the finishing vats, and that means it is unpasteurized. It is felt by almost all beer drinkers that unpasteurized beer has the freshest, most natural taste, but being unpasteurized, it must be kept cold (40 degrees or lower), or the still-live yeast in the beer will continue to multiply, ruining the beer. Bottled and canned beer is pasteurized, eliminating that problem, but at the cost of somewhat altered taste. Stout is a dark, strong flavored beer. Stout can be purchased at most bars as a draught beer. So can lager, bitter, porter, ale, bock, dunkel, pilsner, cream or any other type of beer. And all of them can be bottled or canned, too, after pasteurization. Draught is a way of dispensing beer. Stout is a type of beer. That's the difference.
They are not contrasting things. There are 2 types of beer. Lager and ale. All draft beer is, is beer served from a keg thorough a tap. You can serve lager or ale from a tap.