You have completed a task and now you want to mark it as such. Well there are two ways that you are able to do this; from within the contact record or from within the task section - you can pick which way is easier and most convenient for you.
Within the Contact record
Find that contact (via Contacts > Quick Search) or if you are already in the contact you will see the Outstanding Tasks in the top-right of the record:
So here by the outstanding tasks all you have to do is click the tick to mark it as done (you can also edit it if needs be - just a quick tip).
Once you have completed that task you will see it appears in the Interactions section of the Contact record...
Within the task section
Go to Tasks > Search or Find your Task and click on the Tick to mark it as complete:
You can then view your Completed tasks using the Complete filter: