I'd like to set a badge on the icon of my app like in apple's mail app (number on top of the icon).
How can I do this in Swift (iOS8)?
The "number on top of the icon" is called a Badge. Badges can be set on a number of things besides Application Icons including Navigation Bar toolbar icons.
There are many ways to change the Application Icon Badge. Most use cases involve setting this when the Application is in the background to alert the user that there is some change that they may be interested in. This would involve a push notification.
However, you can also change it while your app is active. You will need permission from the user by registering the UserNotificationType. Once you get permission, you can change it to whatever number you wish.
application.registerUserNotificationSettings(UIUserNotificationSettings(forTypes: UIUserNotificationType.Sound | UIUserNotificationType.Alert | UIUserNotificationType.Badge, categories: nil )) application.applicationIconBadgeNumber = 5