If you need to restore an activity that has previously been marked complete within your database, there are a couple of ways to accomplish this.
If the activity was not attached to a contact record, you would need to restore it from your main Calendar. You'll first need to go to the Day or Week view of the Calendar that contains the date when the activity was marked complete. Then, in the Actions widget to the right of your calendar, click Show Completed Activities:
All completed activities within the date range you are looking at will then be on display. As an example, let's say we wanted to restore an activity called "Social policy review" (in the screenshot below):
Clicking a completed activity will bring up a Viewing Completed Activity dialog:
To restore it, simply click the Restore Activity button and then confirm you want to restore it.
If the activity you want to restore was attached to a contact record, you can restore it in the manner above. Alternatively, you can go to a contact record's Completed Activities area to do so.