With tomorrow's Anniversary Update for Windows 10, Cortana becomes the Windows-Internal search engine with Online access, regardless if the user wants it or not.
Getting rid of Cortana
According to this article you can still disable Cortana by editing some values in the registry:
To disable Cortana for the machine: Set the DWORD value "AllowCortana" in the path "HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Microsoft\Windows\Windows Search" to "0".
In case you want to disable Cortana just for the current user, use HKEY_CURRENT_USER instead of HKEY_LOCAL_MACHINE.
I've not yet tried it, since I'm still waiting for the Update. In case you don't want to mess up with the registry, you can download the Regkey-Files on the article page.
In a nutshell
In the Registry-Path
Set the DWORD Value "AllowCortana" to 0 do disable it for the whole machine. Use HKEY_CURRENT_USER instead of HKEY_LOCAL_MACHINE if you want to disable it for your user only.
Keywords: Windows 10, Disable Cortana, Anniversary Update