I was just looking at a question and I clicked the little star underneath the vote and then I realized that I had no idea what it did.
It marks the question as a favourite.
If you hover your cursor over the star (assuming you’re using a device that has a cursor to hover), you’ll see the description:
This is a favourite question (click again to undo)
All this means is that you found the question interesting, and want to follow it. It has no impact on reputation. You can see all your favourited questions on the /favorites tab of your user profile. Many people use favourites as an informal bookmarking system.
There’s an FAQ on Meta.SE that explains, in detail, how favourites work and what they mean.