Self-diffusion refers to diffusion processes in the absence of a concentration gradient or chemical potential gradient. This diffusion is essentially random in its direction. If there is a gradient, then particles will tend to move from high to low concentration (often called chemical diffusion). The diffusion coefficients measured by PFG NMR or by QELS / DLS light scattering are self-diffusion. Chemical diffusion is more complex since the gradient makes a difference.

Sometimes a self-diffusion coefficents is measured in a pure liquid (water for example), other times in a solution (water in various solvents). The diffusion coefficient for any material dissolved in a medium will depend (at least) on both the viscosity of the medium, other characteristics of the medium and the interaction with the material (polarity, etc), and on temperature.

There are many different types of diffusion coefficients which depend on which diffusion formalism you consider, if under "diffusion coefficient" you mean chemical diffusivity coefficient in mixtures (e.g. mixture averaged models, Stefan-Maxwell diffusivity, multicomponent diffusion coefficients for different kinetic models).

For gases it depends on the gas parameters (e.g. temperature, pressure, composition) and transport parameters (e.g. molecules cross-sections) in a complicated manner. It is derived from the similar provisions as self-diffusion coefficient (e.g. kinetic-theory) but can't be associated with it directly by any relation.