How do I calculate pi (Using Gregory Series) in Java with input dialog
boxes?  I need to prompt the user for the number of terms to estimate
pi.

I would recommed writing a seperate backend class to handle the PI
computation.  Have a method take all the necessary inputs then return
PI.

If you just need popups for inputs, use JOptionPane.showInputDialog()
and JOptionPane.showMessageDialog()
[http://java.sun.com/j2se/1.4.2/docs/api/javax/swing/JOptionPane.html
< api] methods for input and response.  Write a simple program to use
these to feed inputs into the backend class and return the response.

If you need something more complex, the frontend using the Swing
library.  If you  don't know how,
http://java.sun.com/docs/books/tutorial/uiswing/ is a good resource.