Next to Planning, Rules, Conversations and Activity a light grey number says how many unread updates the page has (99+ above 99, nothing at 0), and pointing at the tab says it in words, for example 3 unread updates. Next to Conversations it counts the conversations with new messages from others; until version 13.2.49.77 it counted the conversations waiting for a reply from you or your agents, which the Conversations page still shows with its Waiting filter. On those pages the unread plans, rules and conversations have their title in bold, like unread mail, and an unread Activity entry looks like an unread conversation: its title in bold with a small green dot, on a light green background.
An update is unread when someone else, another team member or their agent, or your own agent created or changed it after you last read it; only what you do yourself in the app is never unread for you. In Conversations only the conversations you are part of count, and a new message makes a conversation unread. Opening a row in the panel on the right marks it read and the number goes down at once, and Mark all as read, in the ... menu next to the project's name, marks read everything the open page shows (all four pages, on the Overview); it is there only when there is something to mark.
The numbers and the bold rows follow the team's work within a few seconds while the project is open. SyncNow keeps what you have read on its server, for you and for each project, so it is the same on every PC. The first time SyncNow counts a project for you, when you open it or when it starts following a project you have never opened, everything older counts as read, so you are not flooded with bold rows. In Projects, the badge next to the project's name is the sum of the four numbers, the same total you see split across the tabs once you open the project; a change to a plan or a rule is also an Activity entry, so it counts once in its page and once in Activity. On the Windows taskbar, SyncNow's button shows a small green badge with the total of what you have not read in your projects, 1 to 9 or 9+, and it disappears when everything is read; when SyncNow is hidden in the notification area the same number is drawn on its icon there. A project whose Show notifications is unticked in Project settings, under Notifications, is not counted there, but keeps its badge in Projects. Windows does not show badges when the taskbar uses small buttons. See What does a project row in the Projects list show?.