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  
  

This is where most basic CAD modellers would stop. The problem with just putting a cylindrical or a spherical fillet in a corner is that you basically go from a flat surface (zero curvature) to a surface with some curvature on a whim. If your surface is reflective that means you go from a flat mirror to a strongly distorting one instantly, this will visually appear as a edge even if there is none. Curvature btw. is just the reciprocal of radius (1/r)

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.  
  

Now the curvature of a curve can be drawn as a curvature comb. You basically take the curvature at any point of the curve and draw the value as the length of a line that is perpendicular to the curve.

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.

  position
  velocity
  acceleration
  jerk
  snap
  crackle
  pop