What is meant by isotropic diffusion?
As Olli has said, Isotropic means the same in all directions. So the diffusion behaves the same regardless of which direction it is diffusing in. 

Heat for example diffuses in an isotropic manner. (Well ... unless you have a strange environment with varying conductivity constants or something ...  but that's another story)

The heat equation is

$\frac{\partial u} {\partial t} = k \nabla ^2 u $

Where $U(x,y)$ is heat at point $(x,y)$ and $t$ is time

$k$ is the thermal diffusion constant.

I think it helps to see visually what this looks like



(From wikipedia)

This is actually just like doing Gaussian Blur on an image.

(Gaussian convolution is a solution to the heat equation)

Gaussian blur before and after

Note that the whole image gets the same blur 

Anisotropic diffusion on the other hand may diffuse differently in different directions. 

For example, perona and malik suggested doing diffusion based on the Laplacian (that's the 'edginess' $\nabla^2$) of that part of the image. 

This can be used for performing blur only in certain parts of an image. 

See here for an image which has had anisotropic diffusion. This has had the effect of removing noise, while preserving the main features of the image.

The equation proposed by Perona and Mailk is

$\frac{\partial U}{\partial t} = c(x,y,t) \nabla ^2 U + \nabla c . \nabla U$

where $c $ is a function such as

$C(||\nabla U||) = e^{-(||\nabla U||/K)^2}$ for example