First, I need to download the <ahref="http://drupal.org/project/date"title="Date module on Drupal.org">Date</a> module, and make my module dependent on date_popup by adding the following line into my module's .info file.
dependencies[] = date_popup
Within my form builder function:
~~~php
$form['date'] = array(
'#title' => t('Arrival date'),
// Provided by the date_popup module
'#type' => 'date_popup',
// Uses the PHP date() format - http://php.net/manual/en/function.date.php
'#date_format' => 'j F Y',
// Limits the year range to the next two upcoming years