Google Chrome desktop notifications which appear for new gmail mail and new chat are a part of google chrome which is enabled by default when you install the latest version of google chrome, just in case if you don’t like it you can easily disable them in google chrome.
Please note that these notifications will appear only when you are running google chrome and has gmail opened so don’t get confused that these notifications are a part of gmail, as these are desktop notifications so you can disable them through google chrome, in my case I disabled them as I use google talk all the time and I get all the new email and chat notifications from google talk so I don’t need these desktop notifications which were anyways duplicate for me.
You can easily disable these notifications by selecting Disable these notifications from mail.google.com as shown in the image above, alternatively you can go to chrome settings – options >> under the hood >> Privacy >> Content Settings
Under Content Settings >> Notifications – Manage Exceptions – here you will see the mail.google.com with allow behavior just click the cross symbol to remove this entry from here and all the desktop google chrome notifications will be gone, they will no longer appear on your desktop as shown in the image below.
One more tip which could be useful for you, in case you are annoyed because of the position of these desktop notifications, you can also change their position as shown in the snapshot below.