Why does attraction always cause release of energy making the system more stable? Suppose $\ce{Na^+}$ and $\ce{Cl^-}$ form an ionic bond. In doing so they will come closer by attraction. Now as they come closer their speed will increase and hence the kinetic energy increases. So basically the electrostatic potential energy will be controverted into kinetic energy.


  
  My question is why do these processes that involve attraction cause a release of energy from the system? And after this release why is the system said to be more stable?
  


My understanding is there might be two concepts involved. 

First, as the ions approach each other their speed increase, but as soon as they touch each other they collide and reflect back. So the ions keep on oscillating. These oscillations are what we know as heat and the temperature of $\ce{NaCl}$ will be higher than its surroundings so it will release its energy in the form of heat.

Second, it might be because any accelerating charge releases electromagnetic waves, and so as long as the ions oscillate they'll keep releasing energy in the form of EM waves which may be trapped by other ions or may escape.

After the energy will be released from the system(NaCl ion here) our system must become more stable. Stable in the sense that NaCl crystal will not convert back into ionic form separated from other ions because for coming back to that state the system must be provided the lost energy back by some external agency. This is the way how I understand the term stabilty; I don't what does this term mean in chemistry. I also don't know why " every system in nature wants to be more stable" -- This is the phrasing that I heard in school days from chemistry teacher. This fact must be related to some thermodynamic fact of kind that every system wants to release energy, but why?
Attraction is a shorthand way of saying "potential energy gets lower as the distance gets lower".  The opposite is true of repulsion.  The potential energy of two like-charged particles, e.g. two $\ce{Na+}$ atoms, will increase as they are brought closer together.  

Attraction (and its opposite, repulsion), are forces with units (in SI) of Newtons, or $\frac{\rm kg\cdot m}{\rm s^2}= \rm N$.  

Energy has units of Joules, or $\frac{\rm kg\cdot m^2}{\rm s^2} = {\rm N\cdot m} $.

The relationship between a force $F$ between two particles separated by a distance $r$ and the potential energy $E$ of those particles is $F = -\frac{dE}{dr}$.  

If the energy gets lower ($dE$ is negative) when two particles get closer together ($dr$ is negative), then the force $F$ is attractive. If the energy gets higher ($dE$ is positive) as the distance gets lower ($dr$ is negative), then the force is repulsive.  

Force and energy are two different mathematical ways of looking at the same thing.  That's why they are always related.$%edit$