** Step 1) A Copyright Holder, or "an agent authorized to act on a copyright-holder's behalf" files a copyright complaint against a video.
** Step 2) Youtube removes the video (allegedly after a short check, to prevent Unauthorized Copyright Claim Spam).
At this point, either...
** Step 3a) ...the ACTUAL copyright-holder can submit a counter-notification, which says "Sorry dudes, that claim was bullshit and that person can have their content back." Which rarely happens because the copyright holder stands to gain nothing, unless someone was trolling them by having them remove their own content
OR
** Step 3b) The person whom the claim was filed against submits a Claim Dispute, providing appropriate legal documentation (such as the Creative Commons License) or other certifications or proof of Fair Use (letters from Hanna-Barbara and Time Warner which legally allow you to invoke the trademarked name "Scooby Doo" on-camera, or a letter from Carlos Santana's Record Label that allows you to not have your videos audio deleted because you forgot to turn off the radio during your Vlog, etc.)
** Step 4) After this, Youtube makes a final decision internally, on their own authority rather than a legal one. Additionally, if they still disagree (which they did in my case by discarding my Claims Dispute and apparently not even reading it), there can be no more Claims Disputes unless you hire a lawyer and file a lawsuit against Youtube. And after claims are contested, Youtube has a nasty habit of letting the person who filed the Copyright Claim against you start monitizing your videos, as happened here:
http://www.rebelliouspixels.com/2013/buffy-vs-edward-remix-unfairly-removed-by-lionsgate