How do I make a rectangle move across the screen with key bindings?

1. You should override paintComponent in your JPanel and call super.paintComponent(g) in it.
2. See How to Use Key Bindings tutorial. Key bindings are preffered in this case rather than a KeyListener
3. pack() then setVisible()
4. You should set global variables for x anf y location, so they can be accessed from within your Action. Then in your actions, increment your x or y and repaint

Try running this example

import java.awt.*;
import java.awt.event.*;
import javax.swing.*;

public class KeyBidings extends JFrame {
    int x = 0;
    int y = 0;

    DrawPanel drawPanel = new DrawPanel();

    public KeyBidings(){
        Action rightAction = new AbstractAction(){
            public void actionPerformed(ActionEvent e) {
                x +=10;
                drawPanel.repaint();
            }
        };

            InputMap inputMap = drawPanel.getInputMap(JPanel.WHEN_IN_FOCUSED_WINDOW);
            ActionMap actionMap = drawPanel.getActionMap();

        inputMap.put(KeyStroke.getKeyStroke("RIGHT"), "rightAction");
        actionMap.put("rightAction", rightAction);

        add(drawPanel);

        pack();
        setDefaultCloseOperation(EXIT_ON_CLOSE);
        setLocationRelativeTo(null);
        setVisible(true);
    }

    private class DrawPanel extends JPanel {


        protected void paintComponent(Graphics g) {
            super.paintComponent(g);
            g.setColor(Color.GRAY);
                    g.fillRect(0, 0, getWidth(), getHeight());
            g.setColor(Color.GREEN);
            g.fillRect(x, y, 50, 50);
        }

        public Dimension getPreferredSize() {
            return new Dimension(400, 200);
        }
    }

    public static void main(String[] args) {
        EventQueue.invokeLater(new Runnable(){
            public void run(){
                new KeyBidings();
            }
        });
    }
}

Here’s the code you’re more concerned with

    Action rightAction = new AbstractAction(){
        public void actionPerformed(ActionEvent e) {
            x +=10;
            drawPanel.repaint();
        }
    };

    InputMap inputMap = drawPanel.getInputMap(JPanel.WHEN_IN_FOCUSED_WINDOW);
    ActionMap actionMap = drawPanel.getActionMap();

    inputMap.put(KeyStroke.getKeyStroke("RIGHT"), "rightAction");
    actionMap.put("rightAction", rightAction);

Create a custom action and add that action to the action map, linked to the input map keystroke. In the action, just increment or, decrement the x and/or y, depending on the direction, then repaint the panel.

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See Key binding tutorial | Graphics tutorial