Drupal: How to Render Results of Form on Same Page as Form

The key information in the link Nikit posted is $form_state['rebuild']. Here's some info from Drupal 7 documentation that I believe applies the same for Drupal 6...

$form_state['rebuild']: Normally, after the entire form processing is completed and submit handlers ran, a form is considered to be done and drupal_redirect_form() will redirect the user to a new page using a GET request (so a browser refresh does not re-submit the form). However, if 'rebuild' has been set to TRUE, then a new copy of the form is immediately built and sent to the browser; instead of a redirect. This is used for multi-step forms, such as wizards and confirmation forms. Also, if a form validation handler has set 'rebuild' to TRUE and a validation error occurred, then the form is rebuilt prior to being returned, enabling form elements to be altered, as appropriate to the particular validation error.