For questions on Meta Stack Exchange about the tag watching feature (formerly known as favorite tags), which allows a user to select a set of tags on a Stack Exchange site that will be highlighted in question lists.
Tag watching (formerly known as favorite tags, or interesting tags) allows a user to select a set of tags on a Stack Exchange site. All questions with those tags are highlighted in the questions page. If a user is looking at a tag search page for a watched tag, the only questions highlighted will be those that have other watched tags.
There's a list of all watched tags in the Tag watching section of the settings in your user profile and on the right-hand side of every questions list page, along with an entry field to set them. They can also be set by hovering over the tag, and clicking on the "Watch tag" button. You can unwatch by clicking on the same button (though the label will have changed to "Unwatch tag").
You can also ignore certain tags by clicking on "Ignore tag". Questions with ignored tags can be set to hidden from question lists completely, or just faded out. (For questions on Meta Stack Exchange about the ignored tags feature, see the ignored-tags tag.)
The watched/ignored tags are used by the Interesting Tab of the home page in Stack Overflow only as part of its algorithm to determine which questions to show.
Using the search operator
intags:mine, you can restrict any search to your watched tags.
Read about the announcement for Tag Watching.