Push notifications were introduced by Apple in one of the iOS releases a few years back - iOS 4 I think. Apple introduced it as a way of improving performance on iOS devices, and as a way of standardising the way that alerts are delivered to the apps that you have running on your device. If Apple didn't implement push notifications as a 'standard' way of generating alerts, each app would have to find its own way of doing this which would result in apps that were more 'buggy', more confusing to configure, and that would probably have a bad impact on battery life.
The way it works is that a developer working on an app that needs to tell a user when something has new happened (such as a new Facebook 'like', a new Whatsapp message, or a new email) can register his app with Apple's push notification service. When the new event happens, (for example, you got a new email) the developer's 'back end' system pushes an message not to your device directly, but to Apple's push notification service saying "MacLija got a new email". The push notification service then delivers the "New email" alert to your device, and you get the little red dot on the app icon. This means it's not the Facebook app (or whatever) that actually produces the alert on your device - it's Apple' Push Notification system that actually does that.
When you install an app to your iPhone which is capable of sending push notifications, you normally get a message when the app installs: "<app name> would like to send you Push Notifications", and you can choose "OK" or "Don't Allow" at that point. You can choose to mirror these notification settings (so your Watch will produce alerts in the same way as your phone), or not by opening the Apple Watch app on your phone, tapping Notifications, and then enabling or disabling push notifications for each app with the sider.
When the settings are configured, any push notification that is sent to your phone while your phone is locked and not in use, is then pushed by your phone on to your Watch.
If your phone is in use, then the phone displays the push notification and does not push it on to your Watch as it assumes you have just seen the alert on your phone.
Fella.