Tokyo Toshokan
Before I list specific kinds of anime download sites, I felt I should mention one of the best general download sites, Tokyo Toshokan. It's literally updated every few minutes or so with people uploading torrent files from a variety of anime fansubs, mp3s, j-pop soundtracks, manga scanlations, video clips, and tons of other things! It's definitely the first place to check out if you're looking to download anime stuff ^_^
Bittorrent Fansub sites
Get the most recent episodes, movies, and music of Pokemon in English, Japanese, and subtitled!
AnimeYume.org
(another Anime Yume, but this one's ".org" not ".com" ^^,,,)
Kaizoku Fansubs
See One Piece in all its orginal, unedited Japanese glory at Kaizoku Fansubs =)
Anime Destiny
Fansubbed episodes of Konjiki no Gashu Bell(a.k.a. Zatch Bell) available here!
Dattebayo Fansubs
Get your Naruto, Bleach, and other fansubs here!
Mp3, video clip, and direct download sites
Anime Kitchen specializes in anime mp3s, schedules, and reviews!
AnimeMusicVideos.org has a database of thousands of great fan-made anime music videos!
Keiichi Anime Forever has many different kinds of anime downloads, but probably its best feature is that it has hundreds of anime episodes available for direct viewing! ^_^
Gendou's Anime Music
Gendou's has one of the best selections of anime mp3s!
Galbadia Hotel has a ton of anime and game mp3s, midis, and art books available for download
This site has a ton of great anime episodes available for download(however, it also has some adult/hentai episodes as well, so be warned...)
Manga download sites
Lurk Manga Archive
If you know how to use IRC, Lurk is the place to go for manga. They literally have thousands of manga scanlations available from hundreds of different series'
One Manga
The best source for easy, online manga reading with hundreds of titles to choose from!
Manga Volume
Choose from over 200 series' and read manga directly online at Manga Volume!
Stop Tazmo
Download a variety of manga directly from this site
AD Inuyasha
Get the latest scanlations of the Inuyasha manga here!
Freelance-Manga
Freelance-Manga also has scans of the Inuyasha manga