Is it just going downhill fast on a pair of planks?
Downhill skiing is a form of snow sport that involves majority of the time skiing downhill with some speed, hence of its namesake. Most people participate in the sport of downhill skiing at ski resorts "lifted", which are chairlifts that are designed to carry snow slope users from the bottom of the run back up to the top of the run. There are however are some time when going up hill is required at "lifted" resorts due to slope placements or the needs of the skier. Downhill skiing is sometimes called Alpine Skiing, particularly in contradistinction to Nordic skiing, which is cross country skiing and ski jumping.
The telemark turn is the oldest form of turning on skis. It was superseded in the 1930s, when christiana turns were developed. The parallel turn is a form of christie. Christies only really became possible when heels were locked down onto the ski. The telemark was revived in the 70s in America, and has a small but devoted (and growing) band of supporters.
The telemark turn involves bending the leading knee and pushing it forward. The heel on the trailing ski is lifted, and that knee is dropped. The leading ski goes forward, and both skis form one long curved edge, which is ridden to the end of the turn when the leading leg and trailing leg swap, and the process is repeated.
Initially leather boots were used, but in the last 15 or so yyears plastic boots have been developed. Telemark bindings have been mounted on downhill skis and the combination has meant that telemarking has been taken to steadily more extreme terrain.
Watching a good telemarker in action is a thing of beauty.
Cross Country (XC skiing)
Cross Country skiing would have to be one of the original form of skiing in the early days. Cross Country skiing involves the skiers travelling through the bush on a pair of skis. Ski touring involves travelling as far as you want on cross country skis. Many people spend several days on their journey and camp out in the snow. Cross country skiing is closer to bushwalking than to downhill skiing in its experience and ethos, although going downhill on skinny skis with free heels has its own particular excitement.
In the Alps of Europe, a type of alpine bindings were developed (and technological advance continues) that allow the skier to unlatch the heel binding for uphill travel, using "skins" for grip on the snow. At the top of the ascent (e.g. a col or a peak) the skier strips off the skins and latches down the heel bindings to allow normal alpine (downhill) skiing. These suit steeper terrain (e.g. ski mountaineering), as opposed to the gentle terrain suited to XC skiing.
It is quite possible to use Alpine Touring (AT) bindings for normal ski area skiing, and keep skins in a day pack for a quick jaunt out back, in order to access snow where the lifts don't go.
There are a wide range of competitive things done on snow from downhill alpine skiing, to XC to aerial spinning, to sliding very fast down a big slippery dip to shooting things and a whole range of other things.