An isotropic material is one which looks the same in every direction. We cannot define any special direction using the material properties. In other words, none of the properties depend the orientation; it is perfectly rotationally symmetric. Note that in order to be isotropic the material must be homogenous on the length scale of interest, ie the same at every point in the material.
For instance, rubber is a very isotropic material. Take a rubber ball, and it will feel the same and bounce the same however you rotate it. On the other hand, wood is an anisotropic material: hit it with an axe and it will take more force to break of you are cutting across the grain than along it. (Remember we're thinking about the material rather than the shape of the object.)