I've been active on MO for a while now, but I have not yet figured out the role of the star that appears on the left-hand side of a question, underneath the number up/down votes. Sometimes there are numbers underneath this star. I clicked on some of these stars, the numbers changed but I still don't know what it all means, and I have not been able to find an explanation on the MO Help site. Thanks.
2$\begingroup$ It marks the question as one of your favorites, and then the question shows up in your favorites folder (found on your activity page). $\endgroup$– LuciaNov 7, 2015 at 21:33
$\begingroup$ Placing the cursor on the star shows up the explanatory text "This is a favorite question (click again to undo)". Then you can search the internet for "mathoverflow favorite question". This gives a bunch of information. $\endgroup$– Todd LeasonNov 8, 2015 at 10:03
$\begingroup$ Here is another related post: What's the purpose of the favorite button? $\endgroup$– Martin SleziakAug 21, 2018 at 10:14
Clicking the star makes this question a "favorite" of yours, clicking again undoes it (in contrast to votes you can change this at your will indefinitely).
The meaning of "favorite" is as in some browsers, to "bookmark" or "follow" the question.
Once marked it will show up in the "favorites"-tab on your user page. This allows you to relocate a question easily. And not only this, you also see a notification (a small blue box with a number in the tab) to signal new activities on the threads that are your favorites.
Note that one can also see other users favorites. Moreover, if a question is a favorite of many users, its poster will get a badge ("Favorite Question" for 25 and "Stellar Question" for 100).
For more complete information see How do favorite questions work? on the general meta-site.