How do I make my tour private or take it off Explore?

It depends on how the tour is published, and all of it is on the tour's Publish tab. Only the tour's owner can change it.

• To take it off Explore, untick Publish to Global Showcase on the Overview pane and click Save Settings. A tour that is not hosted is then private again: visitors get Tour not published, and only you and your collaborators can open it. A hosted tour with Published ticked still opens by its link.
• To make a hosted tour private, untick Published on the Hosting pane and save. It is then like a draft for everyone else.
• To keep a hosted tour to the people you choose, leave it published and choose under Who can open it on the Hosting pane: only the people you send its link to, or only the people who know its password.
• To pause it for a while, tick Unpublish / Hide Tour on the Hosting pane: visitors, on Explore or on another website, see Tour paused until you untick it.

A tour that is not hosted cannot be kept to its link: without hosting, the only way to publish a tour is to list it on Explore. Who can see a tour has every state.