I am new to differential geometry and I am trying to understand Gaussian curvature. The definitions found at Wikipedia and Wolfram sites are too mathematical. Is there any intuitive way to understand Gaussian curvature? 
For a intuitive understanding, imagine a flat sheet of paper (or just grab one in your hand).  It has zero Gaussian curvature.  If you take that sheet and bend it or roll it up into a tube or twist it into a cone, its Gaussian curvature stays zero.

Indeed, since paper isn't particularly elastic, pretty much anything you can do to the sheet that still lets you flatten it back into a flat sheet without wrinkles or tears will preserve its Gaussian curvature.

Now take that sheet and wrap it over a sphere.  You'll notice that you have to wrinkle the sheet, especially around the edges, to make it conform to the sphere's surface.  That's because a sphere has positive Gaussian curvature, and so the circumference of a circle drawn on a sphere is less than $\pi$ times its diameter.  The wrinkles on the paper are where you have to fold it to get rid of that excess circumference.

Similarly, if you tried to wrap the sheet of paper over a saddle-shaped surface, you'd find that you would have to tear it (or crumple it in the middle) to make it lie on the surface.  That's because, on a surface with negative Gaussian curvature, the circumference of a circle is longer than $\pi$ times its diameter, and so, to make a flat sheet lie along such a surface, you either have to tear it to increase the circumference, or wrinkle it in the middle to reduce the radius.

Indeed, in nature, plants can produce curved or wrinkled leaves simply by altering the rate at which the edges of the leaf grow as compared to the center, which alters the Gaussian curvature of the resulting surface, as in this picture of ornamental kale:

$\hspace{170px}$

For more nice illustrations, see for example these two articles.
