In which method of Action class the business logic is executed ?

In the execute() method of Action class the business logic is executed.

public ActionForward execute(
   ActionMapping mapping,
             ActionForm form,
             HttpServletRequest request,
             HttpServletResponse response)
          throws Exception ;

execute() method of Action class:
Perform the processing required to deal with this request
Update the server-side objects (Scope variables) that will be used to create the next page of the user interface
Return an appropriate ActionForward object