You have to use:

 getAppletContext().showDocument
     (new URL("http://www.whatever.com"));

or

 getAppletContext().showDocument
     (new URL("http://www.whatever.com"),"HTML frame ID");

If "HTML frame ID" do not exists then a new browser window will be opened. The following "HTML frame ID" have special meanings :

   "_self"    current frame
   "_parent"  parent frame
   "_top"     base frame
   "_blank"   new window

For example, we want to display lowres.html page if resolution is 640x480 else the hires.html is used.

 import java.applet.*;    
import java.awt.*;
import java.net.*;

public class whatres extends Applet {
  public void init() {
    Dimension dim = Toolkit.getDefaultToolkit().getScreenSize();
    if (dim.width==640 && dim.height==480) {
       try {
         getAppletContext().showDocument
           (new URL(getCodeBase()+"lowres.html"),"_top");
         }
       catch (Exception ex) {}
      }
    else {
       try {
         getAppletContext().showDocument
           (new URL(getCodeBase()+"hires.html"),"_top");
         }
       catch (Exception ex) {}
      }
  }
}


In the example below, you can type a new URL in a textfield, and press a button to go to that page.

import java.applet.*;
import java.awt.*;
import java.awt.event.*;
import java.net.*;

public class GotoURLButton extends Applet implements
    ActionListener {
  Button b;
  TextField t;
  
  public void init() {
     t = new TextField(20);
     add(t);
     b = new Button("Go to this URL");
     add(b);
     b.addActionListener(this);
     }
  
  public void actionPerformed(ActionEvent ae) {
     if (ae.getSource() == b) {
       try {
         getAppletContext().showDocument(new URL(t.getText()));
         }
       catch (Exception e) {
         e.printStackTrace();
         }
       }
     }
  }