A counter-notice is your formal answer to a takedown. It says, under penalty of perjury, that your video was taken down by mistake or because the material was misidentified.
The right comes from the DMCA (17 U.S.C. § 512(g)). When we receive a complete counter-notice, we must send it to the rights holder and restore your video 10 to 14 business days later, unless the rights holder goes to court first.