Gmail notifications in Notification Center
I assume you are talking about a Mac with OS X?

To make the Notifications stay till you dismiss them for emails received in Mail.app:

Open the system preferences + Notification.

Click on Mail.

Choose the "Alerts" (not the Banners).

It will stay displayed till you click Close. After that it is no longer in the Notification center.

This is not the same as the original Gmail notifications. To get those:

You need to be signed in to Gmail and have Gmail open in your browser to see desktop notifications. If you want to use that, then you need to use Chrome browser with a extension called "Checker Plus for Gmail ". With that extension you can configure the duration and other stuff.

If you want to filter the Notifications, you can use the mail.app Rules.

For example for Junk mail make a rule to mark it as read, and it wont show in the notifications.



You can play with the Mail-Rules for other emails you do not want to see in the Notification (set them as read).