What is the difference between regulator and stabilization What is the difference between regulator and stabilization in control theory
don't they both minimize the disturbance to the system?
could answer be elaborated from the view of state and output?
 A: There is a difference between the ability to stabilize a system, and an ability to control it. 
Stability implies that the states and outputs of a system do not grow unbounded. A good discussion of stability can be found in the book: "Nonlinear Systems" by Hassan K. Khalil. 
For autonomous dynamical systems (i.e. systems with no inputs: $\dot{x}=f(x)$) stability means that if the initial conditions of the system states, $x(0)$, lie near an equilibrium, then they will stay near that equilibrium.  
For dynamical systems with inputs $u$, and outputs $y$ (e.g. $\dot{x}=f(x,u)$, $y=h(x,u)$), input-output stability means that for inputs of bounded magnitude, the outputs of the system will also remain bounded in magnitude. Internal stability implies the additional requirement that the states will also remain bounded.
Based on this definition of stability, it is clear that it is impossible to control the outputs of a system unless it can be stabilized. However, it should also be clear that stability alone doesn't provide any guarantees that the outputs of the system can be manipulated in a controlled manner, for instance to negate the effects of a disturbance. 
Control systems that are designed to manipulate the outputs of a dynamical system, are in effect manipulating the system input to: (a) stabilize the system, and (b) make the system output behave in a desired way. The design of such systems can be broadly divided into two categories (this concise definition is from the book "Multivariable Feedback Control" by Skogestad and Postlethwaite):


*

*Regulator problem: Manipulate the system input to counteract the effects of disturbances.

*Servo Problem (i.e. tracking problem): Manipulate the system input to keep the output close to a given reference trajectory, e.g. minimize the difference between the output and the reference trajectory.
A: If I understand your question properly, we have two things to consider.
Stability
This is the ability for a system to operate under a variety of conditions without self destructing. There are two categories of interest; 1) the ability for the system to return to equilibrium after an initial displacement away from equilibrium and, 2) the ability for the system to produce a bounded output for a bounded input. There are differences between time-varying and nonlinear systems.
Regulator
In this system the reference input is zero (since it is missing). It is desired to keep the output as near to zero as possible in the system.
So, in both cases, we certainly want them to remain stable, but one allows two types of inputs, while the other wants to have a constant zero input. One can ask why they defined two different concepts for this and that is a valid question, but they needed a system that didn't have any inputs as opposed to the general case I suppose.
