Activities can be archived in one of 2 ways: using the mass management controls in Activities Builder or from the activity build page. 
When you archive an activity, it will be removed from Activities Builder and put into the Archive. 
Any existing assignments of that activity will be deleted. The activity will also be removed from any feature where it was previously appearing (e.g. Self Assign, Record Completions, etc.) and be unavailable for assignment in Create New Assignment. Completion records of archived activities will be available in Reports and they will still show up in the user’s My History.
A caveat to the above description is Credentials. If you have an archived activity that is part of a credential, that activity will still be there and any completions will still count for credit, but the activity will show as "archived" in both Manage Credentials and on the user’s credential detail page. And no new completions of that activity will be allowed, regardless of credential topic settings. 
Quick Glance
Archived activities:
- All assignments of the activity are deleted
- It is removed from all features where it could be completed or assigned
- It can still be reported on
- It still appears in credentials, but cannot be completed
- Completions still count toward credential credit
- Completions still appear in user's My History