From this answer to How do comments work?
When should comments be deleted?
Comments are temporary "Post-It" notes left on a question or answer. You should not expect them to be around forever: Once a clarification has been made, an edit added to the post to include new information, or the issue in the comment is otherwise resolved, it is subject to deletion. In reality, many obsolete or chatty comments remain untouched due to the high volume of comments posted, but this does not mean that they can't or shouldn't be deleted in the future.
Who can delete comments?
A user may delete one of his own comments by clicking on the (x) icon that appears to the right of the comment's timestamp when the comment is moused over. A user can delete only one comment every five seconds.
Comments that are flagged by multiple users are deleted automatically. The number of flags needed is based on the comment's score. It currently takes "3 + (Score / 3)" flags to delete a comment.
Moderators can delete any comment, or purge all comments from a post.