Every Time you create an anonymous class, it retains an implicit reference to its parent class. So when you write:

public class LeakyActivity extends Activity
{

...

    foo.registerCallback(new BarCallback() 
    {            
        @Override
        public void onBar() 
        {
            // do something                
        }            
    });
}

You are in fact sending a reference to your LeakyActivity instance to foo. When the user navigates away from your LeakyActivity, this reference can prevent the LeakyActivity instance from being garbage collected. This is a serious leak as activities hold a reference to their entire view hierarchy and are therefore rather large objects in memory.

How to avoid this leak:

You can of course avoid using anonymous callbacks in activities entirely. You can also unregister all of your callbacks with respect to the activity lifecycle. like so:

public class NonLeakyActivity extends Activity
{
    private final BarCallback mBarCallback = new BarCallback() 
    {            
        @Override
        public void onBar() 
        {
            // do something                
        }            
    });

    @Override
    protected void onResume()
    {
        super.onResume();
        foo.registerCallback(mBarCallback);
    }

    @Override
    protected void onPause()
    {
        super.onPause();
        foo.unregisterCallback(mBarCallback);
    }
}