== does plain reference comparison. It is an operator.
equals() is a method defined in java.lang.Object class. Subclasses can override this method to provide a meaningful content comparison. The default behaviour implemented by the Object class does nothing but reference comparison. So, if a class doesnot override equals() method, the result of calling this method will be consistent with the results obtained when you use the == operator.
CompareTo() is a method defined by the java.lang.Comparable interface. Unlike the equals() method, this method is designed to be used for determining the natural order of objects while sorting.
Java provides various sorting helper classes and method and some of them call the compareTo() method( if implemented by your class ) to findout the natural sorting order of each object that is an instance of your class. So, compareTo() does not check equality, but checks which object comes first when they need to be sorted in a specific order.
Ajith