I am planning to implement Socket.io in android by this library for a chat based application. As far as I understood the library seems to be pretty good. I want to know how
First of all, the Application's onCreate() is irrelevant in your use case because you can't have a thread running in the background when it was first initiated in a non service code.
Also, I would recommend using Google Cloud Messaging instead of creating your own mechanism. It would be best for the device's battery life and much less code for you to handle.
If you do want to implement that chat completely on your own, Service is your only choice. You can also combine it with a singleton, but I wouldn't recommend that approach. You can use broadcasts and BroadcastReceiver for communicating between your Service and Activity, I think it is easier than a Bound Service since bounding to the service is asynchronous and it creates a lot of mess compared to simple broadcasting.