If you were researching for something at office before heading out to a meeting with your handset. You had to remember and re do those searches at Google Search.
But that is history! as Google introduces Personalized Suggest for Android, iPhone and Palm WebOS.
So what ever you searched from your desktop is only a thumb click away on your mobile top. You open Google.com on your mobile and click the search box and select the historical search from the drop down.
But make sure that you are using the same Google Account on both the devices.
Official Google Mobile Blog: Your mobile and desktop search history now in sync