Physical Interpretation of the Diffusion Constant $k$ I have read technical explanations of the interpretation: 

''Diffusion coefficient is the proportionality factor D in Fick's law (see Diffusion) by which the mass of a substance dM diffusing in time dt through the surface dF normal to the diffusion direction is proportional to the concentration gradient grad c of this substance: dM = −D grad c dF dt. Hence, physically, the diffusion coefficient implies that the mass of the substance diffuses through a unit surface in a unit time at a concentration gradient of unity.''

Is this the only definition of a diffusion? The definition of the word means spreading out of something, to diffuse. What does it mean to ''diffuse through a surface.'' 
 A: Possibly the confusion arises because "through a surface" can be interpreted as "along (or within) a surface" or "perpendicular to a surface". The second case applies here. The surface is taken to be one whose normal vector is parallel to the particle movement:

It's true that diffusion can be viewed as a dispersion of matter from a point source; here, an alternative view (that 1-D diffusion tends to erase a 1-D gradient) is taken to easily quantify the flux.
A: The simplest version of Fick's law is
$$J =- D\frac{\partial \phi}{\partial x} .$$
This is described in detail at the Wiki page, but in a nutshell, the equation tells us that as the concentration gradient increases, the "diffusion flux" or flow of particles per unit volume and time (J) gets larger. 
The diffusion constant D depends on the squared velocity of diffusing particles, which in turn may depend on other properties such as viscosity of fluid and size of diffusing particles. 
What is the physical meaning of D?
Suppose that $\frac{\partial\phi}{\partial x}$ is high (we have a steep gradient exerting a strong force on particles). If D is very small (perhaps because of physical properties of the fluid) this will inhibit flow despite the gradient. If the gradient is quite small but D is large, we could have the opposite effect. For a given gradient, increasing D will increase flow. 
If the gradient is the engine driving diffusion, D governs the degree to which that engine will be engaged to give the resultant flow J. 
What does it mean to "diffuse through a surface?" It means to cross from one side of a surface to the other under the influence of an osmotic gradient. In general the surface may be a mathematical construct or a physical surface whixh allows or partly allows the passage of particles.
A: There are some good answers already. I want to give a different yet simpler understanding of the diffusion constant by looking at its units. 
The (SI) units of the diffusion constant are given by 
$$[D]=\frac{\rm m^2}{\rm s}$$
any this is true for diffusion in any number of dimensions. People sometimes naively think that this represents an "effective area" that particles will diffuse throughout in a given time period, but this interpretation doesn't make sense, for example, in 1D diffusion. 
What the $\rm m^2$ really represents here is the mean squared displacement of a diffusing particle. So a larger diffusion constant means that you would expect diffusing particles to "spread out more" since they are moving out to larger distances over time.
