A double bevel adds a second, more acute, angle behind the edge bevel. This secondary bevel is generally called a back bevel or relief angle. Its purpose is to thin the metal behind the edge. The thinner the edge, the greater the cutting ability. However, an edge that is too thin is susceptible to damage. By adding a smaller, more obtuse primary bevel to the very edge, it gives the edge the strength to avoid damage from impaction, chipping or rolling. In addition, as a knife is sharpened over time. . . .the metal behind the edge becomes progressively thicker. As a result, the knife doesn’t cut as well and becomes harder and harder to sharpen. The answer is to grind the shoulders off the edge at an acute angle (i.e., add a back bevel, then reestablish the primary bevel).
For a better understanding. . . . .look at blade #5 in the pic below.
A micro bevel is a very tiny bevel at the very edge of a knife. The principle reason for a micro bevel is to provide toughness right at the edge, without sacrificing the geometry of the blade.