Chatter is a key social feature of the platform; it can enable users of your application to collaborate and communicate contextually around the records in your application as well as optionally inviting their customers to do so, using the Chatter Communities feature. It is a powerful aspect of the platform but covering its details is outside the scope of this book.
You can enable Chatter under the Chatter Settings page under Setup, after which you can enable Feed Tracking (also under Setup) for your Custom Objects. This setting can be packaged, though it is not upgradable and can be disabled by the subscriber org administrator. Be careful when packaging references to Chatter such as this, as well as including references to the various Chatter-related objects, since this will place a packaging install dependency on your package, requiring all your customers...
