public void methodName(String results) {
    this.results = results;
}

Button.addActionListener(new ActionListener() {
          public void actionPerformed(ActionEvent event) {
                    this.methodName(asdf);
}

Something some = new Something() {

    public void overridden() {
        YourClass.this.methodName("test");
    }

};

Button.addActionListener(new ActionListener() {
    public void actionPerformed(ActionEvent event) {
        methodName(asdf);
}

final MyClass enclosingClass = this;
Button.addActionListener(new ActionListener() {
    public void actionPerformed(ActionEvent event) {
        enclosingClass.methodName(asdf);
}