Under the pointer they turn green, with no tooltip and no light green square behind them (since version 13.2.49.77, like the arrow on the right); clicking the rest of the row opens the project.
A small green dot next to the project's name means someone else is working on the project right now: one of their agent sessions is open on it, or they changed something in the last 5 minutes; your own sessions never light it. Pointing at the dot says Someone is working on this project. The dot goes away by itself when nobody else is working.
To turn it off for a project, open Project settings (the Settings tab, or Project settings in the ... menu next to the project's name) and untick Show notifications, under Notifications; the project then also stops notifying its new posts and stops counting in the number on SyncNow's taskbar button, while the grey number of unread updates next to its name stays. Show notifications applies to everyone who manages the project, on every PC. See What does the Activity tab show?.