I'm sure there's probably some anime enthusiasts that would groan at me for putting this show on my list of favorites, but tough cookies. This show is the reason I got into anime in the first place. InuYasha first aired in the U.S. in 2002 on Adult Swim, and since that was around the time I started college, I watched it a lot (probably when I should have been studying). The series was an English-dubbed version, which makes me cringe a bit now (I like reading subtitles).
The story follows Kagome, a young high school girl who falls into a well and ends up in the Sengoku Era of Japan. There she meets InuYasha, a half-demon, and they embark upon various adventures trying to recover the pieces of the powerful Shikon jewel. You can read more about the storyline here.
In my opinion, this is a "classic" anime that any fan should watch. It's not the deepest, most involved storyline, but it's entertaining and fun, and that is what's most important to me in an anime. You can also watch all 7 seasons online for free on Hulu. It's one of those shows that you don't really have to pay close attention to- meaning you can crochet while watching it. Multi-tasking is awesome.