Could the uniqueeInstance have one and only one instance?
public class A {
     private static A uniqueInstance = new A();
     private A() {}
     public static A getInstance() {
            return uniqueInstance;
     }
}
Could the uniqueeInstance have one and only one instance?
public class A {
     private static A uniqueInstance = new A();
     private A() {}
     public static A getInstance() {
            return uniqueInstance;
     }
}
This is a Singleton pattern, it's purpose is to have only one possible instance for a class.
This is why you have a private constructor , so that no other class can attempt to instantiate it directly.
Here are more elaborate thoughts for possible uses of a Singleton :
It is not guaranteed.
By reflection you are easy to get more instances.
The only way to guarantee that you have exactly one instance is to use an enum:
enum Holder {
    INSTANCE;
    //Keep in Mind of A you may still have more instances. if you want to have the
    //guarantee to have only one instance you may need merge the whole class
    //into an enum (which may not be possible)
    public A uniqueInstance = new A();    
}
Other ways like throwing an exception in the constructor are generally also possible. But not completely secure since there are ways to create an Object without calling any constructor.