How should I implement object comparisons in a flexible manner? For example, I have a Person class and sometimes I will compare based on name and sometimes I will compare based on age.

Instead of having the Person class implement the Comparable interface, you could delegate the comparing to another class. Perhaps you could have a PersonComparator interface that you could implement for the various types of comparisons. For example:

public interface Person {
  public String getName();  
  public int getAge();
}

public interface PersonComparator {
  public int compare(Person p1, Person p2);
}

public class AgeComparator implements PersonComparator {
  public int compare(Person p1, Person p2) {
    if (p1.getAge() == p2.getAge()) return 0;
    return p1.getAge() > p2.getAge() ? 1 : -1;    
  }
}

public class NameComparator implements PersonComparator {
  public int compare(Person p1, Person p2) {
    return p1.getName().compareTo(p2.getName());
  }
}