OK, all is clear now! The normal position of the search in the header is above the logo and to the right on the same line as the Home & Login bar. Your template moves navMainWrapper to below the logo, and I presume the navMainSearch is turned off in admin. The code for it may have been deleted from tpl_header.php.
The search you see is not in the header at all, but in the right sidebar (navColumnTwo). You can turn that one off in Tools > Layout Boxes Controller > sideboxes/search.php and turn on sideboxes/search_header.php. If the search doesn't appear where you want it, add the code back in as shown above so there is something for the switch to control.
Someone definitely did a bunch of custom template work, as a stock ZC install will not look or behave as yours does. Find your /template_default/ directory. There will also be a /classic/ directory, and there should be another in the same place which has your template files. If the edits are in /template_default/, you could be in big trouble when you upgrade to the next version of ZC.



