In snowboarding and presumably skiing, carving is the act of going back and forth as you go down a hill, to decrease your speed so you dont end up a pile of broken bones at the bottom of the hill, to increase control, and to turn. The tighter the turns, the more you will slow down (in theory). I'm not sure about the mechanisms of skiing, but in snowboarding the key is to catch the uphill edge of your board in the snow and drag it somewhat perpendicular to the direction you are moving. If properly done, his results in snow spraying impressively from the downhill side of the board, and a reduction in speed. To do this properly, you continually zig-zag back and forth, which means you must carve both toeside and heelside. This is much easier said than done, especially on steep slopes.

The biggest cause of crashes while attempting to carve is allowing the downhill side of the board to dig into the snow, which will flip you downhill, and send you sliding through the snow face first. It looks impressive, but isn't too fun. It is important to keep as much weight as you can on the uphill side of your board. Also, when turning, it is important to put almost all your weight forward and kick the back end of the board around. (this is counterintuitive to some people especially since in skateboarding it's often the other way around) This is especially vital on steep slopes because if you don't make the transition between toeside and heelside fast enough, you will be pointed straight down the hill and will accelerate to intense speeds. This can be fun if you're an expert snowboarder, but not if you're trying to learn to carve.

I recommend learning on some of the less radical 'intermediate' slopes. Most 'bunny' slopes are not steep enough to carve properly on.