If you click on the "gray" star next to the title of a thread, it turns gold and is bookmarked for you. You can see all your bookmarked threads at the top next to your username, inbox, etc.
Gold stars just indicate that you're following a thread and will get notifications in it when someone posts. You can also track this with the "Participated" tab above, if you don't want your inbox to get slammed.
Grey stars are just... well, you're not doing anything about it, but you can click on it to turn it gold.