What are enums and why are they useful?

Question

Today I was browsing through some questions on this site and I found a mention of an enum being used in singleton pattern about purported thread safety benefits

Answer 1

It is useful to know that enums are just like the other classes with Constant fields and a private constructor.

For example,

public enum Weekday
{
  MONDAY, TUESDAY, WEDNESDAY, THURSDAY, FRIDAY, SATURDAY, SUNDAY
} 

The compiler compiles it as follows;

class Weekday extends Enum
{
  public static final Weekday MONDAY  = new Weekday( "MONDAY",   0 );
  public static final Weekday TUESDAY = new Weekday( "TUESDAY ", 1 );
  public static final Weekday WEDNESDAY= new Weekday( "WEDNESDAY", 2 );
  public static final Weekday THURSDAY= new Weekday( "THURSDAY", 3 );
  public static final Weekday FRIDAY= new Weekday( "FRIDAY", 4 );
  public static final Weekday SATURDAY= new Weekday( "SATURDAY", 5 );
  public static final Weekday SUNDAY= new Weekday( "SUNDAY", 6 );

  private Weekday( String s, int i )
  {
    super( s, i );
  }

  // other methods...
}