To understand EM waves, you need to start from Maxwell's equations.
Here's what it basically says:
An electric field that varies with time results in a magnetic field that varies with time.
A magnetic field that varies with time results in an electric field that varies with time.
Lets say you have an antenna. (just a long piece of wire) You connect an oscillator to it. So, electrons move back and forth in it. This induces a magnetic field in free space, which in turn results in an electric field and so forth.
Both Electric and Magnetic fields are in phase. i.e., they rise and fall together, but they are perpendicular to each other, and both are perpendicular to the direction of propagation of the wave.
This is called a TEM wave.
The TE, TM waves all exist only in waveguides. (hollow metal stuff or an optical fiber) (In optical fibers, the same thing is called with a different nomenclature)
TE- No electric field in direction of propagation.
TM- No magnetic field in the direction of propagation.
There's one more mode, which is the hybrid mode, in which there's both electric and magnetic field in the direction of propagation.
Basically, these modes exist only because of the boundary conditions imposed by the metallic walls of the waveguide.