A key component of OS X nowadays is push notifications. For example, if somebody shares a photo stream with you then a notification will appear on your Mac to tell you. This has been “pushed” to your computer.
If you’ve reason to believe push notifications aren’t working properly, or if you’re on a different network compared to usual and simply want to ensure push notifications will work on it, you can use the free of charge Push Diagnostics to test. Just click the link at the bottom of the program window to start the testing process, which will take a few seconds to complete.