Without being too technical, curvature of field, as far as I understand it, is the tendency for a lens not to focus all the elements at a given subject distance at the film plane. If you think in terms of the human eye, our eyes focus light on a curved surface, the retina at the back of our eye. Lenses with significant curvature of field are acting like they have a curved film plane instead of a flat plane on which they are focusing the light. I once had a Leica 50/1.5 Summarit that had significant curvature of field. The center of the image would be in focus while the image got progressively unsharp as you moved from the center. As you stopped down, you could see the image get increasingly sharper across the frame as depth of field increased at the film plane. Therefore, "improved curvature of field" should probably be read as the lens does not exhibit very much curvature of field.
Resolution "on axis" would be how many lines per milimeter a lens can resolve at the center of the film, where light is passing, for the most part, straight through the lens. Off axis would refer to the image area as you move away from the center of the film towards the edges/corners. Light striking these areas are entering and exiting the lens at an angle. Most lenses suffer at least some loss of resolving power as you move "off axis."
I hope this helps,
Kevin
