Completing a Reminder will mark it as being resolved and remove it from the To-Do lists, but it will stay on the Calendar as a Completed item. It is different from 'Archiving' a Reminder, which will remove it from the calendar and you will not be able to add it back.
To complete a Reminder, select the Dashboard tab and click on the Reminder in the Calendar or in one of the To-Do items. Then set the Completed Date (by default it will show today's date) and click 'Mark as Complete'.
This will change the button to Mark as Incomplete.
If you are finished, you can now close the window.
To reset the Reminder back to it's previous status of Incomplete, you can click the Mark as Incomplete button.