You can search while signed out of your Google account, but they can still save your history by using browser cookies. For a more private search, use "private browsing" or "incognito mode" (Chrome's term for it).
Most browsers have this choice in the "File" menu: New Incognito Window or New Private Window.
In a private window, the appearance will change to indicate you're in a private window (examples below). When you're browsing in this mode, there will be no cookies set, no browser or search history stored, and no information saved that you've entered in forms. This is useful when you are logging on to a private account from a public computer (or the computer of a friend).
(In Safari private windows, only the search bar turns black).
It's important to remember, that even in private mode, you are not completely private. Google reminds you of this on their help screens.
According to Google,
"Your activity might still be visible to:
This is because websites keep logs of the IP addresses of computers that visit them, and people responsible for your school's or employer's network also keep logs. Your internet service provider also keeps these logs. (In the U.S. ISPs don't need to get your permission to track you in this way. Many other countries have stronger privacy protections).
To keep your activity private from these sources, you'll need to use other tools, such as a VPN or TOR, discussed below.
Here are a few other tools that you can use to protect your privacy.