Chapter 12 in Classical Electromagnetism by Jerrold Franklin (copyright 2005), deals with waveguides.
Waveguides are hollow metal transmission lines for electromagnetic (EM) energy, propagated in the form of EM waves. Waveguides can be any shape, though Franklin discusses the simple cases of cylindrical, rectangular, and coaxial waveguides. Waveguides cannot propagate just any EM wave. Instead they are only useful for EM waves that have wavelengths similar to or shorter than the cross-sectional dimensions to the waveguide itself. (Doing a quick check using a calculation done in homework showed that the cutoff frequency for a rectangular waveguide with dimensions 4 mm x 6 mm did indeed correspond to an electromagnetic wave with a wavelength of 6.65 mm. Only EM waves with higher frequencies and shorter wavelengths can propagate along the waveguide). (Image source: Wikipedia, http://en.wikipedia.org/wiki/Waveguide)
According to the website, All About Circuits, waveguides are important because, at microwave frequencies (100 MHz to 300 GHz), energy does not transport well through wires or lines. They "suffer from large, parasitic power losses due to conductor “skin” and dielectric effects."
When discussing how EM waves propagate through waveguides, we talk about the waves in terms of their orientation with respect to the direction of propagation along the waveguide. There are three transmission modes that we study: Transverse Electric and Magnetic waves (TEM), Transverse Electric waves (TE), and Transverse Magnetic waves (TM). Transverse means that no part of the field is in the direction of propagation.
These different modes occur due to the boundary conditions imposed by the conductive walls of the waveguide. At a conducting surface, there can be no tangential electric field (i.e. there can be no component of the electric field parallel to the walls on the waveguide, only perpendicular to them). Similarly, at a conducting surface, there can be no component of the magnetic field perpendicular to the surface (the magnetic field can only be parallel to the walls of the waveguide).
n × E = Eparallel = 0
n · B = Bperp = 0
In transverse electric and magnetic waves, both the electric field and the magnetic field are perpendicular to the direction of propagation of the wave.