How can I view or restore completed activities?

When within a contact record, under the Contact Overview menu click Activities:

By default you will see the contact's open activities, but you can click "show completed" on this page to view their completed activities:

You'll then see something like the below:

You can then click the Subject Line of the Completed Activity to bring up its Activity Details: 

After clicking the Subject Line, one of the areas on the Activity Details page for the activity is titled Actions: 

Click "Uncomplete" within this area and your activity will once again be set to Open.

You can also accomplish all of the above from your main Calendar. Just use the Actions menu above your Calendar to Show Completed Activities, click the completed activity you want to restore and follow the same instructions as above.