"This is a favorite question (click again to undo)"
is what you get if you position the mouse pointer on that star that appears besides each question.
What effect does that star have?
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It lets you track the question in your favorites tab. In this case, updates to the question (answers, etc) will increment the red counter above the tab (the counter isn't visible if there have been no changes since your last visit), and the updated question will be highlighted yellow.
Effectively, it's a way of "watching" the question.
You can also use it to bookmark questions.
There also is a page where you can view your network-wide favorites
These pages are both tabs on your site and netw0rk-wide profiles. You can get to your network profile by clicking the "network profile" link (it has the SE logo next to it) on your site profile.