Open your Chrome browser and click on the hotdogs in the top right corner (that's the three little bars). Then choose Settings.
Close the settings window and navigate to Gmail. Click the Gear icon in the top right corner and choose Settings. Find the Desktop Notifications section and turn on Gmail notifications (two options).
*Note: You must be signed in to Gmail and have Gmail open in your browser to receive desktop notifications.
That's it. Now you'll be notified when a new email arrives.