Anime music videos (AMVs) have become a rather common part of anime fandom, thanks to computer video editing suites and video-sharing websites such as YouTube. The basic idea of an anime music video is that an anime fan will edit together shots from their favorite anime series and set them to music. Some anime conventions even have AMV contests. Many different kinds of anime are used for anime music videos; you can see anime music videos for such series as Pokemon, Fruits Basket, Naruto, Hamtaro, Speed Racer, Cowboy Bebop, Bleach, as well as many, many others.
However, these anime music videos fall into a "gray area" of copyright law. For starters, the anime fans who are compiling these videos generally have not received permission from the copyright holders to use the footage they are using to make their videos. Also, more often than not, these anime fans also use music by well-known recording artists; generally, they also have not received permission from the copyright holders to use the music in this manner.
I consider this a "gray area" of copyright law, because some people who make anime music videos simply make them for their own enjoyment, and not distributing them anywhere. In this case, the making of the anime music video could be argued as being a case of "fair use." However, the vast majority of people who make anime music videos post them on video sharing websites on the Internet. By doing this, however, these individuals have increased the risk of being found out by a copyright owner, and having their video removed from the service, or receiving a "cease and desist" letter from a copyright holder.
If you decide to watch an anime music video, be warned that there are varying levels of quality. There are some very well-made and creative anime music videos that are available to watch on the Internet. However, there are also quite a few that are either "cheesy" or are just simply not well-made. In some cases, the person making the anime music video may decide to try to make a lot of literal matches between the footage and the music; as an example, any time water is mentioned in the lyrics, footage of water is used at that point. This kind of editing is a form of "cheesy" anime music video.
In the end, it's up to the viewer to decide whether or not they enjoy anime music videos. It is also up to the individual to decide whether or not anime music videos should be an integral part of their anime fandom.