Assume you have some objects which have several fields they can be compared by:
public class Person {
private String firstName;
private String lastN
If you implement the Comparable interface, you'll want to choose one simple property to order by. This is known as natural ordering. Think of it as the default. It's always used when no specific comparator is supplied. Usually this is name, but your use case may call for something different. You are free to use any number of other Comparators you can supply to various collections APIs to override the natural ordering.
Also note that typically if a.compareTo(b) == 0, then a.equals(b) == true. It's ok if not but there are side effects to be aware of. See the excellent javadocs on the Comparable interface and you'll find lots of great information on this.