As has, to some extent, been mentioned before, an enum is a java class with the special condition that its definition must start with at least one "enum constant".
Apart from that, and that enums cant can't be extended or used to extend other classes, an enum is a class like any class and you use it by adding methods below the constant definitions:
public enum MySingleton {
INSTANCE;
public void doSomething() { ... }
public synchronized String getSomething() { return something; }
private String something;
}
You access the singleton's methods along these lines:
MySingleton.INSTANCE.doSomething();
String something = MySingleton.INSTANCE.getSomething();
The use of an enum, instead of a class, is, as has been mentioned in other answers, mostly about a thread-safe instantiation of the singleton and a guarantee that it will always only be one copy.
And, perhaps, most importantly, that this behavior is guaranteed by the JVM itself and the Java specification.
Here's a section from the Java specification on how multiple instances of an enum instance is prevented:
An enum type has no instances other than those defined by its enum constants. It is a compile-time error to attempt to explicitly instantiate an enum type. The final clone method in Enum ensures that enum constants can never be cloned, and the special treatment by the serialization mechanism ensures that duplicate instances are never created as a result of deserialization. Reflective instantiation of enum types is prohibited. Together, these four things ensure that no instances of an enum type exist beyond those defined by the enum constants.
Worth noting is that after the instantiation any thread-safety concerns must be handled like in any other class with the synchronized keyword etc.