How do push notifications work? I am wondering how is push notifications system working? 

Is there an active TCP/IP connection that runs in the background to Google servers?
Yes, Android keeps one active connection to Google's servers, but it doesn't use much power or data, because no traffic is sent along it until something sends a GCM message to an app on your phone. There's only one connection on the phone, used by all apps: installing a new app that uses GCM doesn't add any extra load.

The first step in GCM is that a third-party server (such as an email server) sends a request to Google's GCM server. This server then sends the message to your device, through that open connection. The Android system looks at the message to determine which app it's for, and starts that app. The app must have registered with Android to use GCM, and it must have the relevant permission. When the app starts, it might create a notification straight away with the data from the message. GCM messages are very limited in size, so the app might instead open a normal connection to the third-party server to get more information (for example, downloading the headers of new emails).

The advantage of using push notifications is that apps don't have to run at regular intervals to check for new data, saving both power and data. The advantage of having a centralized mechanism like GCM is that the device only needs one open network connection and the Android GCM system is the only thing that needs to keep running, rather than each app having to stay running in the background to keep its own network connection to its own server.