Principles of Wind Power

The Savonius above used drag to rotate itself, the Darrieus uses lift. The blades are shaped across the width rather like an aircraft wing, with the leading edge (facing the wind) thicker than the trailing. They are also curved on one side more than the other. An imbalance of pressure is felt on one side of the blade as the wind passes over, causing a lift rather like that on the wing of an aircraft that keeps it up in the air. Consequently the blade moves away, bringing the other one in line and the process repeats. This means they can in fact spin at a speed faster than the windspeed and so are more suited than the Savonius for turning a generator. The centrifugal forces acting on the blades as they turn are enormous and the blades have to be very strong. Bending them in the manner shown from top to bottom helps to spread the load and gives them their distinctive egg-whisk shape.
Since the blades turn fast, they produce little torque and thus the inertia of the turning assembly cannot be overcome by itself alone- in other words they cannot self-start. One solution is to have an electric motor initialise the turning, another is to put a small Savonius array on to start the turn.
Incidentally this is not, as you would think, a recent invention. The patent was issued in 1931. You can see the disadvantage- they are big, they are ugly and need lots of space to be guyed in to stop from blowing over. (You can see the cables emanating from the top of the tower).

You might stumble on a Darrieus somewhere but something you won't be able to miss is the second type of turbine. I see at least one of them every day. It is:

Horizontal Axis Wind Turbine (HAWT)

Inside it gets even more complicated:

What do all these bits do? Starting rom the nose, pitch control adjusts the angle of the blades to get the best angle of attack on the blade for the wind. The better the angle, the more energy you can extract. The low speed shaft drives the gear box. Underneath that is a brake. It stops the turbine operating by means of a disc on the high speed shaft. The high speed shaft is driven by the output of the gear box. It turns magnets through a coil, producing electricity (in fact it's a dynamo, just like you used to have on bicycles to work the lights and similar in principle to the alternator on a car). The controller takes readings and does all the adjustments automatically, including the yaw which moves it to face the wind. It can be overriden and the turbine can then be adjusted manually. The anemometer measures the wind speed. You'll see why that's important later.The wind vane keeps it stable and the nacelle covers the working parts and saves them from the elements.

If you're thinking that this is high precision mechanical engineering, you are correct. If you're thinking it won't be cheap, you're right again. A very small one for mounting on your house will not see much change out of £600. Installations like Royd Moor, which need specialist construction and transportation, cost millions.