Unless explicitly stated, all YouTube videos (and other online videos) are protected by copyright. Unless the video is licensed for re-use the videos may not be downloaded for use in the classroom unless the educational exemptions of the NZ Copyright Act apply.
However, you may play videos directly from YouTube in a classroom as long as they are part of a lesson, the audience is comprised only of students and teachers from your school, and no one has “paid” to be present when the video is played.
This exemption however does NOT enable teacher or student to re-post the video, or parts of the video (like a mashup) outside the classroom – including online, for example on YouTube or on a blog. This is because the educational exemptions to the Copyright Act do not extend to communicating/showing works publically, unless explicitly permitted to by copyright holders. Such works may only be used within the classroom.
Any work that is used must be acknowledged appropriately to avoid plagiarism.