Are airlines responsible for Airport security or Airports concerned or third-parties on terminal-side? In Indian Airports, all sort of terminal-side security is done by CISF . They do all the frisking and baggage screening etc. I am concerned more for the terminal-side security rather than airside where it comes under the airlines domain. 

Is it similar in the rest of the world or are airlines responsible? Some notable incidents I recall are the recent Brussels Airport attack, and the 2015 Sabiha Gökçen Airport bombing. I would suspect most airports now are under some sort of integrated command and control center modeled on the American system after 9/11 rather than have disparate silo-based security which may have holes that both terrorists and criminals may be able to get through. 

I am talking from port of departure, where the procedure is, show tickets, passport at the main gate, go through security check, get boarding pass and board the plane. 

Similarly, while de-planing, you go through customs and immigration, security check, baggage inspection etc. 

At both places, security check is apparently done. My question is who is responsible, either the Airport Operator or the Government or a multitude of authorities ?

I am trying to be as generic as possible so as to know if there are variances or are most Airports following some set conventions ?
There have been many articles about TSA high-handedness but guess that's to be expected.
In the United States of America passenger and baggage screening is handled by the Transportation Security Administration (TSA). It is a government run agency and deals with the pat downs, X-Ray machines and frisks. This will get you through "security" and to the terminal area and ultimately on to the plane. 

If you are coming in from another country you will need to pass through customs which is managed by its own agents and falls under the jurisdiction of U.S. Customs and Border Patrol who are also a federal agency. 

At some of the really busy airports here in the US its not uncommon to see local police officers helping out with things like traffic flow and passenger management of those entering and exiting the facility and security in general.