Root Locus, Physical meaning of the Roots of the Ch. Equation?

First of all, you should know that root locus method is used to find the values of K (i.e. controller gain) for which a close loop system is stable. read more

Root locus will tell you projectile of the roots wrt a system parameter , so draw the root locus and see whether it stays withing 2 and 4th Quadrants. Nyquist plot is bit more complicated, you need to see are there any encirclement around (-1,0). If there are any then system is unstable. read more

Branches of a root locus represent the path followed by the roots as you keep changing the gain K. Actually root locus can be drawn for any variable parameter of the system, if it affects the solution of the characteristic equation. read more