Euclid: What is the difference between a 'surface' and a plane 'surface'? I've begun to study Euclid's Elements and i've a few questions regarding the difference between a surface and plane surface.

A surface is said to be "that which has length and breadth only", it then goes on to say "a plane surface is a surface which lies evenly with the straight lines on itself". 

What is the difference between the two? What exactly is a plane surface? Can a plane surface be curved, or is it only flat?

I apologise for being a pedant, these definitions should be straight forward but I wish they could be a bit more specific, I want a good understanding of them before I move on.
The first point to make is that Euclid's definitions are not definitions in the modern sense, so that this kind of confusion is common. A surface, for Euclid, is roughly a two-manifold embedded in 3-dimensional space, e.g. the standard 2-sphere or one of its hemispheres, or the graph of certain functions from the plane to the line. A plane surface is just a surface that happens to lie in a plane, e.g. the unit disk. One more modern way to think about "lies evenly with the straight lines on itself" is that the tangent lines to a plane surface lie within the surface, as does not occur in general.
In differential geometry a 2-D surface is like a sphere which has length and breadth, Gauss curvature K is non-zero. It is doubly curved. 

A flat surface has zero K and is a part of a plane or developable to it, at least one of two radii of curvature is infinity.

If you try to flatten a Pringles saddle shaped chip, circumferential lines tend to overlap before they break up. On the other hand when you step on an egg shell crushing it, it splits the boundary circumferentially.

Such differential deformation behavior is at the back of intuitive physical perception of spatial embedding of positive, negative and zero double curvature of surfaces.. while engrossed in the formalism of their mathematical treatment.