As you will probably recall, classical water wheels let the water arrive at a right angle (perpendicular) to the rotational axis (shaft) of the water wheel.
Vertical axis wind turbines (VAWTs as some people call them) are a bit like water wheels in that sense. (Some vertical axis turbine types could actually work with a horizontal axis as well, but they would hardly be able to beat the efficiency of a propeller-type turbine).
The only vertical axis turbine which has ever been manufactured commercially at any volume is the Darrieus machine, named after the French engineer Georges Darrieus who patented the design in 1931. (It was manufactured by the U.S. company FloWind which went bankrupt in 1997). The Darrieus machine is characterised by its C-shaped rotor blades which make it look a bit like an eggbeater. It is normally built with two or three blades.
The basic theoretical advantages of a vertical axis machine are:
1) You may place the generator, gearbox etc. on the ground, and you may not need a tower for the machine.
2) You do not need a yaw mechanism to turn the rotor against the wind.
The basic disadvantages are:
1) Wind speeds are very low close to ground level, so although you may save a tower, your wind speeds will be very low on the lower part of your rotor.
2) The overall efficiency of the vertical axis machines is not impressive.
3) The machine is not self-starting (e.g. a Darrieus machine will need a "push" before it starts. This is only a minor inconvenience for a grid connected turbine, however, since you may use the generator as a motor drawing current from the grid to to start the machine).
4) The machine may need guy wires to hold it up, but guy wires are impractical in heavily farmed areas.
5) Replacing the main bearing for the rotor necessitates removing the rotor on both a horizontal and a vertical axis machine. In the case of the latter, it means tearing the whole machine down. (That is why EOLE 4 in the picture is standing idle).
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