Light travels at its maximum speed in empty space, since it is electromagnetic and requires no material medium. In void, for instance, lets say light has a particular frequency and particular wavelength, given the relationship c = lambda*f. If light enters a denser medium, like water or air, the wavelength will go down while f is maintained, thus the speed of light will become less than c. The index of refraction of light gives the proportion between the new speed v and speed c, n = c/v.
then, we have sound. sound is a wave of compression and it needs to have a physical medium to proceed. this is a reason why there is no sound in space. thus, for materials that can vibrate, as they get denser, i.e. closer packing of molecules, the sound wave speed increases. sound is faster in solids than in liquids in which is it faster than air. in general, the speed of sound if given by the square-root of the bulk modulus divided by the density. actually, this formula is kind of surprising because density is on the bottom, but actually, bulk modulus kind of goes up more as density goes up, so denser materials will have a higher bulk modulus (in general) and thus a higher sound speed.
the question as to what an EM wave is a great one, and one that took the best scientists hundreds of years to figure out. basically, light is made up of photon packets which are individual particles of energy, but which behave like a wave in aggregate. this is the classical view. a more correct view, in quantum, is that light is a sort of probability wave of energy, in which the wave is made up of the mathematical chance of finding a photon at any one location as a particular time.
and, of course, mechanical waves are propagation vibration in some material. more particularly, it is energy being transmitted among particles, causing particle displacement.
