What makes a wave dispersive?

Dispersion of sound in air, with constant temperature and pressure, is very slight, increasing for very short wavelengths, and for very loud noises. Why? Because the rapid sequence of weak compression/decompression steps as the sound propagates are adiabatic, or energy-conserving, for the normal ranges of sound. This leaves the local pressure, temperature and density unchanged.

As a result, the equation for the speed of sound in an ideal gas is $c^2 = \gamma P/\rho$, with $c$ the speed of sound, $\gamma$ is the adiabatic constant for the gas, $P$ the gas pressure, and $\rho$ the gas density; other formulas are equivalent. Note that intensity and frequency do not appear in this equation. Hence sound is non-dispersive over wide ranges, given stable atmospheric conditions.

Sometimes thunder is given as a counter-example, where a variety of sounds are heard following a lightning strike but this is not due to dispersion; rather it is the multiple branches of the pre-strike, the main strike, and the extended distances covered by the lightning, plus, sometimes, echos.

Light is similarly non-dispersive in the ordinary atmosphere, but changes in pressure, temperature, and humidity change that, hence mirages.

Take a look at Griffiths Introduction to Electrodynamics, particularly the section called "The Frequency Dependence of Permittivity".

Dispersion can arise from the constraints, or bound nature, of the constituent particles in a given medium. For the example of optical dispersion in a dielectric medium, we could picture the electrons as bound, damped oscillators to which a passing light wave applies a driving force. These oscillators have a natural frequency and, depending on the closeness of the light wave's frequency to the natural frequency, the response of the system will vary-- for instance, more or less energy will be absorbed from the passing light wave.