If you are talking about the box providing search suggestions, here is how to disable it:
Click the Preferences link by the search box.
Click "Do not provide query suggestions in the search box."
Click "Save Preferences"

Note: This preference is saved in a browser cookie so if you clear out your browser cookies the preference will be reset.

@Antman: You are thinking of a different box. The one that displays searches you have done before that are stored in the browser auto complete database.
I was refering to the box Google just added to their home page that displays searches from Google's server.