Campaigns and ads are all associated with statistics, and need to be preserved even when they're no longer needed. Because of this, campaigns and ads cannot be deleted from the platform, however they can be archived. Archiving a campaign, or ad effectively removes it from view and prevents it from running live.
Clicking the status drop down in the status column will provide the option to either turn the campaign or ad ON, OFF or Archived. Once you select Archived, the campaign or ad will disappear from your dashboard.
You can view your archived campaigns or ads at any time by selecting All from the search toolbar. Keep in mind, the default for this search option is always "All but archived".
This process is the same from the My Ads tab for creatives.