Facebook – Turn off close Facebook friend notification

If they are on your "Close Friends" list then by default you will be notified whenever they post. To turn that off, click on "Close Friends" in the left sidebar (if it is not there then click "More" next to "Lists" and then click "Close Friends"). Then click on "Notifications" at the top and choose "Off". Of course you could also remove the person from your "Close Friends" list.

Groups are the same way. If you do not want to receive notifications for posts to the group, click on the group name, then Notifications at the top of the page and choose "Off".

It looks like Facebook has done another update so everyone gets more notifications. This thread on the Facebook forum has a couple of solutions suggested, this one seems to have the most love:

Go to the top of your screen and click the little globe symobol where you can view your recent notifications. Hover over one of the them on the top left and click the X. It'll ask if you want to turn off "top friend" activity notifications. Click yes. They will still appear in your news feed, you just won't get notifications for it all.

On a side note I think the icons you've circled in the first picture have more to do with identifying the notification than close friends (photos & status get the star, links get the pin).

Updated

Click the Notifications globe at the top of the Facebook window and when the drop-down drops click the Settings link (the little X's next to each notification have been removed recently). In the "What You Get Notified About" section, you can edit and turn off "Close Friends activity"