This is the code for my JPopupMenu and how I added it, it is supposed to respond when I right click the table:

    JMenuItem deleteRows = new JMenuItem("Delete Row");
    popup.add(deleteRows);

    personTable.addMouseListener(new MouseAdapter() {

        @Override
        public void mousePressed(MouseEvent e) {
            if(e.getButton() == MouseEvent.BUTTON3) {
                popup.show(personTable, e.getX(), e.getY());
            }
        }

    });

I am not sure why the popup menu is not appearing when I right-click the table in the application. I would appreciate it if somebody told me what I am doing wrong.

The trigger for a popup is different for different OS's, you can't simply use mousePressed and your certainly shouldn't be using e.getButton() == MouseEvent.BUTTON3

From How to use Menus, Bringing up a PopupMenu

The exact gesture that should bring up a popup menu varies by look and feel. In Microsoft Windows, the user by convention brings up a popup menu by releasing the right mouse button while the cursor is over a component that is popup-enabled. In the Java look and feel, the customary trigger is either pressing the right mouse button (for a popup that goes away when the button is released) or clicking it (for a popup that stays up).

Instead, you should be checking for each of the mouse events, pressed, released and clicked. You should also be using MouseEvent#isPopupTrigger to determine if the event is a popup trigger for the OS.

Having said all that, it would be simpler to just us JComponent#setComponentPopupMenu and let it decide instead

personTable.setComponentPopupMenu(popup);

Runnable example...

import java.awt.Dimension;
import java.awt.EventQueue;
import java.awt.Graphics;
import java.awt.Graphics2D;
import javax.swing.JFrame;
import javax.swing.JMenuItem;
import javax.swing.JPanel;
import javax.swing.JPopupMenu;
import javax.swing.JScrollPane;
import javax.swing.JTable;
import javax.swing.UIManager;
import javax.swing.UnsupportedLookAndFeelException;
import javax.swing.table.DefaultTableModel;

public class Test {

    public static void main(String[] args) {
        new Test();
    }

    public Test() {
        EventQueue.invokeLater(new Runnable() {
            @Override
            public void run() {
                try {
                    UIManager.setLookAndFeel(UIManager.getSystemLookAndFeelClassName());
                } catch (ClassNotFoundException | InstantiationException | IllegalAccessException | UnsupportedLookAndFeelException ex) {
                    ex.printStackTrace();
                }

                DefaultTableModel model = new DefaultTableModel(10, 10);
                JTable table = new JTable(model);

                JMenuItem mi = new JMenuItem("I'll be your menu for today");
                JPopupMenu popup = new JPopupMenu();
                popup.add(mi);

                table.setComponentPopupMenu(popup);

                JFrame frame = new JFrame("Testing");
                frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
                frame.add(new JScrollPane(table));
                frame.pack();
                frame.setLocationRelativeTo(null);
                frame.setVisible(true);
            }
        });
    }

}