These occur typically when referring to spatial derivatives. When one expands spatial derivatives from Taylor Series expansions, one can take a wavenumber approach to looking at errors and how waves propagate.

When you do the wavenumber approach, you find that the error has a real and complex part of the solutions. The complex component controls dissipation, while the real part controls dispersion (or vice versa, can't recall 100%).

To clarify, dissipation is the function of wave dying in magnitude, while dispersion is when the speed of the propagating wave is not calculated with proper accuracy and it "lags" behind the actual function.

Doing wavenumber analysis, you find that central differences, i.e. the classic second-order difference:

Have only a real component to the wavenumber error, so while they are inherently non-dissipative, they do have dispersion. Likewise one-side differences, i.e. the standard first-order backward difference:

is purely dissipative in that it will propagate the wave at the correct speed, but it HIGHLY dissipative. When high-order schemes (see Tam and Webb DRP schemes, etc) are used, many times artificial dissipation is needed to damp spurious waves before they become problems. Schemes with orders of accuracy as high as 9th order are commonly used as artificial dissipation.