How to Move an Email to Inbox in Gmail

When you click "Archive" you get a little yellow bar at the top of the window asking you if you want to undo your action. Click Undo to restore the mail to your inbox.
If you continued browsing around it disappears.

If you really want the "Move to inbox" button you'll have to search for the mail, or find it in the All mail folder.

You can still use the back-button to get to the mail but it seams that for some reason you can no longer move it to the inbox from there. A workaround is to move it to another label (preferably one at the top of your label list), go to that label and then to the mail. Now you can move it to the inbox (in the "Move to" menu). This is only useful if you don't remember the name if the mail (or something else to search for), and if some time has passed so that it won't appear at the top if the All mail folder.

"Primary" just means it's not in one of the other tabs.

After you've checked the messages you want to move, open the Labels menu. Make sure that the pseudo-labels for the inbox tabs (i.e., "Social", "Promotions", "Updates", "Forums") are not checked and Apply the change. (If some messages are in a label and others aren't there'll be a – in the checkbox. Be sure to clear that.)

If necessary you can use the Move to Inbox button (if available) to make sure the messages are in the Inbox. (You may need to re-select your messages.)

Note that this (probably) won't train Google which messages should be auto-sorted to those tabs. To do that you need to drag-and-drop between the tabs. (A pain in the neck if, like me, you're far from accomplishing "Inbox Zero".)