import javax.swing.JOptionPane;

public class Addition
{

    public static void main(String[] args)
    {
        String firstNumber;
        String secondNumber;
        int number1;
        int number2;
        int sum;
        
        firstNumber = JOptionPane.showInputDialog(”First Num“);//出错位置
        secondNumber = JOptionPane.showInputDialog(”Second Num“)//出错位置
        number1 = Integer.parseInt(firstNumber);
        number2 = Integer.parseInt(secondNumber);
sum = number1+number2;

JOptionPane.showMessageDialog
(
        null,
        "The Sum is :"+sum,
        "Results",
        JOptionPane.PLAIN_MESSAGE
        );
System.exit(0);
    }

}

少年，不用IDE的吗？

import javax.swing.*;

public class Addition {

    public static void main(String[] args) {
        String firstNumber;
        String secondNumber;
        int number1;
        int number2;
        int sum;

        firstNumber = JOptionPane.showInputDialog("First Num");// 用的是中文引号，造成语法错误
        secondNumber = JOptionPane.showInputDialog("Second Num");// 另外，这里少了一个分号
        number1 = Integer.parseInt(firstNumber);
        number2 = Integer.parseInt(secondNumber);
        sum = number1 + number2;

        JOptionPane.showMessageDialog(null, "The Sum is :" + sum, "Results",
                JOptionPane.PLAIN_MESSAGE);
        System.exit(0);
    }

}

细心   小助手还是有必要的