Can someone help me with this?

Your Google searches while signed in to Google are stored at Google, not on your desktop. I've found it useful to find search results that I have previously found.

I didn't realize that desktop might refer to a Google desktop which is a personal search on your own computer, where you install google personal desktop, and everything is indexed on your computer. That would not apply across another computer. Haven't seen that in a long time -- I avoided it because such things tie up my computer for very little return.

To learn more about web history

• Web History
• Web History - Accounts Help

To use previous web history to narrow down your searches

google: google search history find within

To delete your google web search history, do a google search (you won't be able to restore your deleted history)

Google: goggle search history delete

Thanks for your reply. What I found was, that searches from my desk top were seen on my IPhone. From your reply, I understand that it is not vice/versa, that my IPhone searches will not be saved or seen on my desktop. Is that correct?

Yes, If by desktop you mean search on/within your own personal computer software that you install on your computer, as opposed to web searches,

Thank you for your expertise, time and help

This is what you seemed to me to be describing, it used to have the word "Personal" in it something like Google Personal Desktop.

Inside Google Desktop: Google Desktop Update

"discontinuing support for Google Desktop, including all of the associated APIs, services, plugins and gadgets. As of September 14, (2011) Google Desktop will no longer be available for download, and existing installations will not be updated to include new features or fixes."

http://googledesktop.blogspot.com/2011/09/google-desktop-update.html