The location of the poles in the complex s plane determines the behaviour of the closed-loop (i.e. with feedback) system. As the poles move up/down the imaginary (y) axis the oscillation (modulating) frequency increases; as the poles move to the left along the real (x) axis, the rate of exponential decay (of the envelope) increases,  as the poles move to the right, the rate of exponential growth increases (unstable).

So the root locus is a plot showing how the poles (and zeros) of the closed-loop transfer function move around the s plane, as some parameter (usually the proportional gain inside our controller) is varied. By looking at the root locus we get a feel for how the behaviour of our closed loop system (i.e. plant+controller with feedback) changes as we vary the parameter. We can use that information to decide how to set the gain of our controller to achieve the desired closed-loop response (e.g. stability & damping)- assuming our plant model is correct. 

Try constructing some points on the root locus by deriving the closed-loop transfer function using a simple proportional (gain-only) controller. For various gain settings, find the poles and zeros of your trn fcn and plot them on the s plane. You get a locus, instead of discrete points, when the gain changes by an infinitesimally small amount.

Maybe the following links are helpful:

http://www.ce.utwente.nl/lecturesjob/mov/CE04_Rootloci.mov

http://www.ce.utwente.nl/lecturesjob/mov/CE07_sPlaneDesign.mov

Simply, roots loci gives solution modes - stable/unstable, periodic/aperiodic, ... - i.e. every single pieces of the system response.

Refer to the link below,it has helpful to your question.

http://electrical4u.com/root-locus-technique-in-control-system-root-locus-plot/

Hugh's answer I think is an enough qualitative answer. :)

As Hugh's answer suggests the root locus is a very powerful tool for tuning controller parameters, the general setup where you need this tool is when you have your plant and your proposed controller that has parameters k1, k2, ... and you would like to decide what values to choose for those parameters to achieve a desired performance.

one such performance criteria is that you want the system to be stable which means you need all the poles of the closed loop system to be in the left side of the complex plane so you will need to see how each of your controller parameters affect the location of the poles and this where the root locus comes in handy since it tells you the location of the poles corresponding to every value of your controller parameters, of course you still limited by physical constrains that prevent your parameters from having very large values or very small values but the root locus can also predict the behavior of the system beyond this limits which might be not that useful or might have some theoretical research value (as example most of the new materials developed in the past few years were to push the physical boundaries of the old materials to obtain an interesting behavior).

Other criteria for the pole placement is the amount of damping and the response speed, root locus is very useful in such situations and when you have the optimum parameter configuration for your controller you can look to other behavior aspects that can't be defined accurately through the root locus and need for example time domain analysis or similar techniques to further tune your control system.

Imaging that you have a system or plant and that system is very very complex, then when you design a controller whatever controller you need to be sure that the poles and roots from this system or plant are in the left axis in the S-plane, far far away of the zero, then rise the  subactuted, underactuteded systems this depend of the response of your plant or system. Again root locus help you to put this poles en roots in the S-plane

The answer to this question is hidden in exploring some history of control.

When mathemeticians modelled physical systems as differential equations (time domain) the control engineers showed interest in solving them (for stability analysis and to find shape of response). This was easy if system is of lower order but tedious to do if system has a higher order.

Then mathemiticians found the answer to it by transforming the time domain differential equations to frequency domain (Laplace transform (transfer function)), In short what they did was "they transformed the differential equations to algebraic equations" which was relatively easy.

Once again if the denominator polynomial of a transfer function is of some higher degree then it was difficult to find roots and hence the stability. So they deviced the Routh Herwitz criterion to find stability.

At the same time the effectiveness of feedback was explored and they tried to investigate where and how the poles of a system will move as a function of some proportional gain, if the loop around that system is closed. One answer to this was "keep increasing gain and find the roots and plot it on the s-plane" But this was not a feasible answer.

So your answer is:

"Root locus determines how a system behaves physically as the gain of the input to the plant is varied from zero to infinity "

" And we use it in controller design because it is systematic, saves energy and time" 

Dear Muhammad Shahid,

We know that, if the poles of a system situated left position of imaginary axis of poler plot of a system then system is in bounded condition. Suppose a system T(s)=1/(s+2)  pole situated in -2, if we inverse Laplace this T(s)  ....>>> then we get c(t)= EXP (-2) that mean bounded. on the other hand, if T(s)=1/(s-2) then ....c(t)= Exp(2) ...system unbounded. Now consider a gain K for that system....then changing the K value from 0 to high value may tense to infinite. and every case you determine the output response c(t). you can get your answer...it is very interesting...you may do this in MATLAB, in command window first you define this transfer function like " sys=tf(num,den) then call SISOTOOL ....or LTIVIEW. .......****Physically we need to make system bounded and stable and properly optimized requirement basis. That mean Pole should be in  left position of imaginary axis. Now for changing the gain the pole (roots) affected. so we need to set proper gain in the system without affecting the stability level of the system. That's why Root locus is necessary on the practical point of view.*******     

Root locus is only a tool the help in the analysis and the design of any system.

The analysis and the design is associated with the time domain performance.

Poles location of any system in the s-plane determine the associated performance in the time domain. at any instant, we can know the natural frequency, damping ration, settling time, rising time, overshoot (if any), etc. 

Root locus is helping us to map graphically as graph all possible locations of the poles within the system on the s-plane. The different locations of the poles are obtained under the effect of gain changes (proportional gain).

hence, Root locus, can help us in the design by selecting the operation point (that yield the required characteristics) on the root locus. if there will be more than two poles in the system, the dominant poles will be of interest.

In the design, we can select any of the point on the locus  that is fulfilling our requirement, if this point is on the locus (then we know its gain).Otherwise, if the operation point is outside the locus, then we can use lead/lag compensators to force the roots of the poles locations to go through the point that we want.

However, the weak point of root locu relies on the need to have the math system model and the design is based on second order approximation.

As I have mentioned, root locus is a tool and you are not using it physically in the system. You are either analyzing your system mathematically through it, or use it as a design too. In the practice, you are not implementing root locus into your system, you are just using the results from it.

The following are the main reasons for using root locus

1) We don't have to find the roots of the closed loop characteristic equation for each value of proportional gain "K" and yet we plot those roots.

2) We associate the desired performance to poles on the s-plane (notice our minds are tuned to performance due to a pole location in the s-plane)

3) At once we get the idea of whether a proportional controller is enough or lead is required or lag is required.

 IN SHORT WE GET A COMPLETE PICTURE OF THE RESPONSE OF A CLOSED LOOP SYSTEM

The value of K (proportional gain) depends on input and output of the system and that K value is dependent on the roots of the characteristic equation. Root locus is just a plotting method or a graph to indicate the path of roots or damping conditions which vary according to the K value. 

In root locus we will find poles and zeros G (s). Pole and zero position is critical to preserve view stability, relative stability, transient reaction, and error analysis. When the device is placed to use stray inductance and capacitance, this affects the position of poles and zeros. In the root locus technique in the control system, we will evaluate root location, movement locus, and associated information. Use this information to comment on device results.