From alerts, I'm assuming this picture represents is a panel, not a sheet (my understanding is that sheets are overlays over already open apps).
I'm looking to create a custom, or just built in 'clean' popup alert (notification). I can't seem to find anything within NSAlert that speaks of customizing the alert - alertType seems like it might but apparently it is for conveying importance.
An example is something like this:
(source: https://i.sstatic.net/WJhV8.jpg)
The NSUserNotificationCenter class is used to present these "user notifications" in the upper-right of the screen:
import AppKit
let note = NSUserNotification()
note.title = "Hi Stack Overflow"
note.subtitle = "How’s it going?"
note.contentImage = NSImage(contentsOfURL: NSURL(string: "http://cdn.sstatic.net/stackoverflow/img/apple-touch-icon.png")!)
NSUserNotificationCenter.defaultUserNotificationCenter().deliverNotification(note)
(The one you see from Gmail is a custom non-standard notification system that Chrome provides. You can probably take advantage of that if you write a Chrome extension, but NSUserNotificationCenter is more normal.)