Why is impulse response of LTI system important
What is special about the output when the input is impulse signal?
  
  How is the system nature or system parameters concluded by observing the response of the system due to impulse input?


The Fourier transform of the impulse response is the transfer function of the system, i.e. a function that outlines the behavior of the system at all frequencies. Since you can, in practice, build any signal in time as a sum of sinusoidal functions (also called informally frequencies, although it is not rigorous), if you know the transfer function then you can predict the behavior of the system for any input signal.