The main difference between alpine touring and alpine skiing is how you get to the top of the mountain --in alpine touring, you'll be relying on your own two feet instead of sitting on a cushy chair lift. Think of the sport as a combination of cross-country skiing and downhill skiing, with a little mountain-climbing thrown in.
Alpine touring equipment is similar to that of downhill skiing, and some people do use the same equipment for both. The biggest difference (and advantage) of alpine touring equipment is that it's typically much lighter and more flexible than the equipment you'd use on groomed trails (also known as "on-piste") -- it's easy to carry and won't sink into fresh snow. The heavier, stiffer, more traditional equipment works better on packed-down resort trails.
You can think of alpine touring skis -- also known as "randonnee" (French for "hike") -- as a blend of cross-country and downhill skis. Like cross-country skis, they leave your heels free for the walking motion required to climb the mountain. But when you're ready to ski, you can secure your heel to the ski with bindings. Alpine touring skis are shorter and wider than most other skis, for easier maneuverability (and so you don't sink in the snow). And alpine touring boots are lighter, more flexible and more comfortable than downhill boots. Alpine ski poles are longer than other poles, because they're also used as walking sticks. But they're made with carbon-fiber material, so they're still very light.
So you're on your skis, ready to skip the lift and hike the mountain (in ski-speak, that's called "earning your turns"). Your next question is probably: How do I hike a mountain in skis? This is where skins come in. They're strips of material that attach to the bottoms of your skis to increase traction during the ascent. In one direction, the skin is rough for a good grip, so you don't slide back down the mountain. In the reverse direction, it's smooth for sliding along the snow. So, when you get to the top, you peel them off, turn them around and restick them in the opposite direction for the ride down the mountain. They used to be made of real animal hide (hence the name), but now they're usually made of moleskin or a synthetic fabric. Skins are cut to fit the ski and are attached with a glue-like substance that allows you to remove and reattach them without leaving a residue.
On tougher parts of the climb -- especially early on, when skis are unusable because of snow melt -- skins might not be enough. So, you can wear snowshoes or attach crampons (spikes) to your ski boots. Crampons are also helpful when climbing especially steep slopes.
While this is the basic equipment for alpine touring, in the next section, we'll take a look at some safety equipment that may come in handy on the mountain.