Basically, you check if an object is an instance of a specific class. You normally use it, when you have a reference or parameter to an object that is of a super class or interface type and need to know whether the actual object has some other type (normally more concrete).

Example:

public void doSomething(Number param) {
  if( param instanceof Double) {
    System.out.println("param is a Double");
  }
  else if( param instanceof Integer) {
    System.out.println("param is an Integer");
  }

  if( param instanceof Comparable) {
    //subclasses of Number like Double etc. implement Comparable
    //other subclasses might not -> you could pass Number instances that don't implement that interface
    System.out.println("param is comparable"); 
  }
}

Note that if you have to use that operator very often it is generally a hint that your design has some flaws. So in a well designed application you should have to use that operator as little as possible (of course there are exceptions to that general rule).