java.awt.Component. Therefore you need to install
a MouseListener on your applet that responds to 
mouse clicked events. In an applets of this nature it's simplest just to 
make the applet itself the MouseListener.
In this case, since you need to store the points clicked in a Vector,
it's most convenient  to use getPoint(). The coordinates are relative to 
the component to which this event is directed, not necessarily
the global coordinate system of the applet (though in this case they're the same thing).
import java.applet.*;
import java.awt.*;
import java.awt.event.*;
import java.util.*;
public class Dots extends Applet implements MouseListener {
  Vector theDots = new Vector();
  public void init() {
    this.addMouseListener(this);
  }
  public void mouseClicked(MouseEvent e) {
   theDots.addElement(e.getPoint());
   this.repaint();
  }
 
  // You have to implement these methods, but they don't need
  // to do anything.
  public void mousePressed(MouseEvent e) {}
  public void mouseReleased(MouseEvent e) {}
  public void mouseEntered(MouseEvent e) {}
  public void mouseExited(MouseEvent e) {}
  // paint the dots
  public void paint(Graphics g) {
    g.setColor(Color.red);
    Enumeration e = theDots.elements();
    while (e.hasMoreElements()) {
      Point p = (Point) e.nextElement();
      g.drawOval(p.x, p.y, 5, 5);
    }
 }
}