Solid state:
Uses transistors or silicon chips in the gain stage(the main component that does the work).
Characterised by speed, definition, clarity, neutrality and linearity.
Disadvantages: sometimes(in some amps) prone to solid state "glare" or shimmer, can sound overly technical or analytical.
Tube:
Uses old style valves for the gain stage.
Characterised by warmth, laid back sound, euphony and midrange, naturality(not neutrality).
Disadvantages: can sound like you're in a bubble, can add too much "warmth" to the sound, can be slow, bass can sometimes be rolled off, tubes are fragile expensive and burn out -unlike transistors which basically have an unlimited life span, tubes must be matched much more strictly than transistors. Some tubes run very high voltages which are dangerous.
Hybrid:
Tries to take the best of tubes and the best of solid state and mix the two in a multi (usually dual) stage setup which can sound really good or can sound pretty average.
Characterised by tube warmth with solid state speed.
Add the tube disadvantages to the Solid state and you have the potential disadvantages of hybrids.