What is hard and soft water and how does the process of dissolution work?
Hard water and soft water differ only in the presence or absence of certain impurities:


Hard water: Water with impurities in the form of minerals containing calcium and magnesium added in.
Soft water: Water with these minerals removed.


One thing to note is that hard water is not the slightest bit dangerous. Studies have been conducted by the World Health Organization and the National Academy of Sciences, and no evidence was found linking hard water to and adverse health effects - or any effects at all!


  By mere filtering using a cloth or a machine, how can we get rid of minute impurities, (which make water hard) which are not even visible to naked eyes?


This is impossible. Many devices have been proposed, but none have been shown to work. Quoting from this article,


  There are a variety of devices that claim to manage hard water scale using primarily magnetic or electrical technology. Manufacturers generally claim the devices utilize energy to alter the behavior of compounds or elements within the water. They do not claim water chemistry is altered. In fact, the hardness of the water before and after treatment is not changed. This complicates assessment of their performance. While protocol has been established to assess the effectiveness of ion exchange water softeners (NSF/ANSI 44), at this time, no recognized agency in the U.S. has established protocol to assess the effectiveness of physical water treatment devices. Therefore, questions remain as to their effectiveness.


Chemicals are the only way to go, unfortunately.

More information on hard water can be found here and here.


  If table salt is dissolved in water, why do we call it a solution? Don't say that a chemical reaction has taken place and some new product say $\ce{NaH2OCl}$ etc. is formed.


In case you're interested, the reaction is
$$\ce{NaCl + H2O \to Na+ + Cl- +H2O}$$

It fits the very definition of a solution:


  a homogeneous mixture composed of only one phase.


$\ce{NaCl}$ is separated into ions that separate uniformly throughout the water.