There's nothing here that will ever call "display()". You could override the paint(Graphics) method of Applet and call it there:
That will get you started.
The next problem you're likely to face is that the dimensions of the board aren't based on those of the applet, so probably you'll only see part of the board. Instead of the sizes being fixed in Board, the Board should get the sizes from the applet, either as constructor arguments, or by being made into a Component itself, added to the applet, which would have its own coordinates.
I tried adding the paint method, tictac.display(g), but nothing appeared on the display. The "dos" window shows a 'NullPointerException' at Board.display line 63. Maybe there's a problem in passing the values to the draw method?