A kink is a location where a curve or surface dramatically changes direction. For example, the corners of a rectangle are kinks. Kinks can also happen at locations where a curve or surface dramatically changes the amount it curves. For example a rounded rectangle has kinks where the line segments turn into arcs.
Like I mentioned in the sample, kinky surfaces can cause problems later on in the modeling process. So by default, Rhino splits up kinky surfaces when they are added to the document.
Regarding degree, there is some good information on this in the Rhino help file - just search for the “ChangeDegree” command.
Degree=1 curves are not the only curves that can have kinks. A full multiplicity knot in the middle of the knot list means there is a place on the curve or surface that can be bent into a sharp kink. To see an example of this, draw a Circle, turn on its control points, and drag one of them. When extruded, this curve will create a Brep with three faces, not one.
Does this help?