In differential geometry one studies the properties of geometric objects, which don't rely on any parametrization.
Behavior of curves, surfaces, and line congruences on the infintely small scale
Lines of curvature, asymptotic lines, geodesic lines, parabolic curve
If one studies surfaces, one looks on certain curves. For example curves, which go from minimal curvature to maximal curvature or curves with no curvature (asymptotic lines).
For special interest in differential geometry are isometric surfaces. This are surfaces which can be wrapped around each other by deformation. For example every cylinder can be unrolled into the plane.
Minimal surfaces are surfaces with vanishing mean curvature on every point. Minimal surfaces have locally minimal area for given bounding curve.
Surfaces of constant GauÃŸ or medial curvature
Helicoids, sliding surfaces, curves and surfaces of constant breadth