You can receive notifications based on various activity on the forum, such as when someone has quoted you in a topic, has tagged you in a post, likes a post you made on the forum, when someone follows you, and etc. The type of notifications you receive on the forum will be based on your own profile settings.
There are a couple of different ways to access your forum notifications. When you have a new notification, you will see a number appear in a bubble just above the Profile button on the forum. The number in the bubble indicates how many new notifications you have, and if you click directly on this bubble, you'll be automatically taken to the notifications page of your profile to view the notification activity. Otherwise if you do not have a new notification and would like to look back on past notifications you have received, visit your profile using the Profile button at the top of the page and then click on the Notifications tab.
The notifications page will show you the reason for the notification, typically accompanied by which member caused the action that you were notified about. Included will be the date and time that the notification took place. Most notifications have links associated with them, allowing you to click to see what exact post was liked by another member, be brought to the post that you were quoted in, and etc.
You can control the types of notifications that you receive by clicking the Edit Profile button on your profile page, and then click the Notifications tab.