As someone who modeled surfaces like this for a living:

  G0 Positional Continuity: The surfaces touch without gap, but there may be a sharp corner. Example: the corners of a cube  
  
  G1 Tangential Continuity: G0 but additionally the surfaces have the same slope (are tangential) at the point where they touch. Example: adding a circular fillet to the corners of a cube  
  

If we talk about forces (e.g. imagine a skateboard ramp) you go flat (no centripetal force) to circular (constant centripetal force) without any transition inbetween. In effect this will feel like a bump that can throw inexperienced skateboarders of their feet.

This means tangential transitions often do not cut it.

  G2 Continuity: In addition to being G0 and G1 you additionally ensure the curvature is the same where both surfaces meet. This usually means instead of going from a flat surface into a circle you go into a curve that starta out flat and then bends slowly into a radius.  
  

G1 is if the perpendicular lines at the ends of the two curves align. G2 is if the curvature comb at the end of the two lines additionally has the same height (indicating the same curvature at the transition point).

G3 is basically just ensuring that the two curvature combs are tangential at the point where they meet. G4 is ensuring that the curvature combs are not only tangential, but have the same curvature. G5 is taking the curvature of the curvature...

By this point you may be able to sense a pattern.

sounds like every step needs one more derivative to be continuous...?