See comments in code 
public enum Seasons{
    summer(0), spring(1), winter(3), fall(4);
    int order;
    Seasons(int order){
        this.order = order;
    }
    //a method that fits the signature of Comparator<T>
    //https://docs.oracle.com/javase/7/docs/api/java/util/Comparator.html
    public static int compare(Seasons s1, Seasons s2) {
        //Box int to an Integer object, and delegate comparison to its
        //comareTo
        return new Integer(s1.order).compareTo(s2.order);
        //or as  DarkRift 3 proposed  
        //return Integer.compare(s1.order, s2.order);
    }
    public static void main(String[] args) {
        //if you just want to check for equality you can do it simply by
        //using the == operator
        System.out.println(summer == summer); //returns true;
        System.out.println(summer == winter); //returns false
        //if you want to compare based on the int property, use a comperator
        System.out.println(compare(summer, summer));//return 0 meaning equal
        System.out.println(compare(summer, winter));//return -1 meaning smaller
        System.out.println(compare(winter,summer));//return 1 meaning bigger
    }
}