Every page in SharePoint has a search box on the right upper side. If you want to hide the SharePoint Search box on one page, add a Content editor webpart with following code:
<style type="text/css">
#SRSB {display:none}
</style>
This works on both SharePoint 2010 and 2007.
Marijn Somers (MVP) has over 14 years experience in the SharePoint world, starting out with SP2007. Over the years the focus has grown to Office 365, with a focus on collaboration and document management. He is a business consultant at Balestra and Principal Content Provider for "Mijn 365 Coach" that offers dutch employee video training. His main work tracks are around user adoption, training and coaching and governance. He is also not afraid to dig deeper in the technicalities with PowerShell, adaptive cards or custom formatting in lists and libraries. You can listen to him on the biweekly "Office 365 Distilled" podcast.