Use desktop notifications to tell the user about something happening while your app is in the background, e.g. a new message arriving. On some desktops, notifications can also have clickable buttons.
Use notifications carefully, because they can easily distract a user who’s trying to focus. The desktop may have controls to disable all notifications from an application, but you could also offer the user more fine-grained controls.
Wrapper libraries like libnotify make it easy to send notifications with a couple of lines of code. If you need to dig deeper, look at the specification linked below.