Popup email notifications let you know when new messages arrive. You'll find them built into many email and they're even in Mac OS, since Mountain Lion.
Some people hate them because they're a distraction from the task at hand when they're trying to concentrate.
Other people love them because it lets you see at a glance when you have a new email and whether it's important, without having to constantly open your email program.
You can use them with webmail too if you use the IMAP protocol to check your email rather than POP3.
Switching to webmail can mean sacrificing a few bells and whistles, but thankfully Google's Gmail lets you receive popup notifications of new emails and chat messages.
How to turn on popups in Gmail
You'll need to be using Google's Chrome browser to enable popup notifications (which Google is calling "desktop notifications").
To enable this feature click the Gear icon at the top right of the Gmail interface and then select Settings.
Scroll down the General tab and you'll find Desktop notifications.
You've separate control over email and chat notifications, along with the ability to limit email notifications to "important" messages. Google decides which of your emails it thinks are important and you can turn this on or off under the Inbox tab in the settings menu.
With desktop notifications enabled, you'll get a popup message in the bottom right corner of the screen when a new email arrives - telling you the sender and subject along with the first line of the body.
Click on the notification and it will open the email in a new browser window. You can also click on the spanner icon to change which corner the popups appear in. If you're using Mountain Lion, you'll find that Gmail automatically taps into Mac OS' new global notification system.
Keep in mind that Gmail's desktop notifications rely on you having the Chrome browser open and being signed into to your account.