Calling a method inside another method in same class

Java implicitly assumes a reference to the current object for methods called like this. So

// Test2.java
public class Test2 {
    public void testMethod() {
        testMethod2();
    }

    // ...
}

Is exactly the same as

// Test2.java
public class Test2 {
    public void testMethod() {
        this.testMethod2();
    }

    // ...
}

I prefer the second version to make more clear what you want to do.

It is not recursion, it is overloading. The two add methods (the one in your snippet, and the one "provided" by ArrayList that you are extending) are not the same method, cause they are declared with different parameters.

The add method that takes a String and a Person is calling a different add method that takes a Position. The one that takes Position is inherited from the ArrayList class.

Since your class Staff extends ArrayList<Position>, it automatically has the add(Position) method. The new add(String, Person) method is one that was written particularly for the Staff class.