slow-wave and fast-wave

Slow-wave structure in that wave velocity is much smaller than the light's velocity (3.10e8). And in contrast, fast-wave structure in that wave velocity can be compared to the light's, I think.

For waves to be fast, their group velocity should be greater than speed of light c. If the group velocity is less than speed of light, it is called slow-wave. By playing with the refractive index of a medium we can make the wave slow, fast or ordinary.

slightly different with what ozgur_io explained, phase velocity, rather than group velocity is used in the definition about slow wave and fast wave.
when it comes to the slow/fast wave structures, most of them belong to traveling wave antennas.
A typical slow wave antenna is the surface wave antenna, which radiates power flow from discontinuities. While leaky-wave antenna is a fast wave structure.
More details could be found in "Antenna Theory"( Balanis,2nd edition)

Group velocity means the velocity of energy flow, which
can not faster than the speed of light; only phase velocity could be faster than the speed of light.

Oh yes that's my terrible mistake. Thanks for correction.

Slow-wave (guided-wave): vp<c or β>ko
Fast-wave (leaky-wave): vp>c or β<ko