Understanding how notifications work on the Apple Watch
The Apple Watch will only mirror notifications for apps having notifications enabled on the companion iPhone. So the first step is to go to Settings > Notifications and choose which app is allowed to display notifications on your iPhone. Once done, open the Apple Watch app, tap on My Watch > Notifications:
Toggle Notifications Indicator on to display a red dot at the top of the watch face each time you have unread notifications. Scroll to the bottom of your screen and select the apps (Calendar, Phone, Messages...) that can forward notifications to your Apple Watch. Toggle Notification Privacy on to hide the details of your notifications (e.g. the content of a text message).
Managing notifications on the Apple Watch
Each time you receive a notification the Apple watch will alert you with a small haptic motion (a small tap on your wrist) or small sound. Simply tap on your watch face to view the notification.
NB: if the Wrist Raise feature is enabled, then you simply have to raise your wrist to view the notification.
Read and reply to a notification: You can use the Digital Crown to scroll through the info and the reply or dismiss the notification.
Dismiss a notification: Swipe down on the current notification to dismiss it (without deleting it).
Delete a notification: Swipe to the left and tap on Clear to delete a notification.
Clear all notifications: Tap and hold on the watch face to initiate the Force Touch feature and then tap on the X button to clear all notifications.
Why notifications are not being displayed on your Apple Watch
Notifications won't be forwarded to your Apple Watch when the screen of the companion iPhone is unlocked and also when the Do Not Disturb mode is enabled.
To turn off Do Not Disturb mode on your iPhone: Go to Settings > Do Not Disturb. Toggle the Manual/Schedule switch off:
To turn off Do Not Disturb mode on your Apple Watch: Swipe upward to display the Settings Glance. A blue crescent moon icon means that Do Not Disturb mode is enabled. Tap once to disable this feature. You can also disable this setting on the Apple Watch app. Go to the My Watch section > Do Not Disturb.
That's it; you now have the basics for enabling and managing notifications covered.