Currently it doesn't look like most of your code will even get called. The method public void init() will get called when you initialize the applet and
public void start() will get called when the applet starts. Your init(Graphics g) doesn't look like it will ever get called the way it is set up now.
Also, using the graphics class directly usually isn't necessary. Unless your teacher told you to do it that way specifically, you should probably have your name set in a Label and add the label to the applet in the same way you do your button. If your teacher did say to use the graphics specifically, you should understand that the entire applet is redrawn when paint(Graphics g) is called. Any direct painting using methods like g.drawstring that is done outside of this method will be overwritten every time the paint(Graphics g) method is called
