Where can one learn about viewership ratings for anime? In the U.S., companies like Nielsen track viewership of television programs. What is the equivalent of this for Japanese television programs (and, in particular, anime), if any?
The closest thing you can probably find to something like Nielsen is Video Research. It's in Japanese (of course) but google translate does pretty well with the anime titles.

Also, there is this site which lists historical ratings that go back to 2000.

As for anime movies, DVD/BD rankings, there's Oricon.