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A hollow metal conductor through which microwaves may be propagated with little attenuation.
Having either rectangular or circular cross-sections, they are used extensively in radar.
A waveguide is a very general term describing any number of devices that can act as a medium for the propagation of electromagnetic waves in a controlled environment.

The following devices act as waveguides:

Microstrips

Coaxial Cable

Rectangular or Circular tubing (the most common sense of the word)

If you want to be overly pedantic, power lines are also waveguides in a strictest sense.

EM waves will propagate through a given waveguide by exciting what are called modes in the guide. These modes are fundamental EM properties of the guide (related to the eigenvalues, actually, when you study waveguides in painful detail). They define how the electric and magnetic fields can be spatially distributed (sine and cosine functions for rectangular guides, bessel functions for circular guides, and so on) inside the guide while the energy passes through. Different modes have what are called cutoff frequencies, or a minimum frequency below which that mode will not propagate through the guide.

Waveguides have what is called a fundamental mode, which is the lowest frequency mode that can be supported by the guide. Most waveguides are designed for a particular range of frequencies such that only the fundamental mode is the one that is propagated. Higher frequencies can excite higher-order modes in the guide, and these travel at different velocities through the guide. This is generally a Bad Thing when trying to keep your signal coherent; so only the fundamental mode is usually the one that is used.

There are some really interesting waveguides that aren't intended to be as such; consider roadway tunnels that go underwater for a long distance. Have you ever listened to the radio and the signal go away? But your cell phone still worked? Cutoff frequency in effect..

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