Take caution not to use the app to sync the same Google account with itself, as that won't accomplish anything in most cases. It only makes sense if there are two separate Google accounts involved, or if the Google contacts on your iPhone are out of sync with your Google contacts on the Internet.
Generally, you should consider syncing your Google account with one of your Apple accounts, such as iCloud.
See this article for more info on how to select an account to sync with Google:
This help article relates to the iPhone app called "Contacts Sync" which can be downloaded here:
Need help? You can email us at email@example.com