Having a look at the contents of the Phone Memory? In my Samsung Galaxy Y mobile phone, I am able to see the contents of my Memory Card but I am not able to see the contents (the files and folders) that are saved in the internal memory of my mobile.

Is it possible to have a look at it only after rooting the mobile phone? Is there any app for doing so?
You would need to root your phone. 

For file managers with root access, you can try Astro or Root Explorer. If the app loads the default folder, keep going up the parent folder until you can no longer move up; that will be the root folder.