Other Tools
Besides the 2D and 3D systems used to create the animated frames and sequences, there are a lot of other important tools we use all the time.

Photoshop and AfterEffects (Adobe, USA)
Photoshop is a tool that no graphics company should be without. Although not much print work is done directly by the anime companies, Photoshop is still the retouching and painting/drawing tool of choice.
AfterEffects is widely used in the anime industry to composite scenes that need effects that are not available in RETAS and Animo. It is often used to enhance scenes that have already been composited and rendered as well. There are a lot of interesting lighting effects that can be done with it. It can be used on rendered 3D sequences as well to add in effects that would be either too hard to control or impossible in 3D.

Debabelizer (Equilibrium, USA)
Debabelizer is also commonly used for batch processing and conversion of files.

Painter (Metacreations, USA)
Painter is used for illustrations, backgrounds and retouching images.

Speed Razor and Adobe Premiere
Speed Razor and Adobe Premiere are used to edit the finished sequences together and output it to tape. Editing animation is pretty simple as the real editing decisions are all done before anything is ever committed to paper—in the storyboarding phase.

Platform Stuff
In any discussion of computer systems the Platform Thing will raise its ugly head, but no discussion is complete without it. The real test of a computer system (hardware and software) for us is in production and those that don't work so well get eliminated pretty quickly. (The needs and usage patterns of the anime industry may be totally different than other industries, so never think that what works for your company will automatically work right for an anime company.) We've been constantly beating on everything we can get access to for years and we will continue to do so, never being happy with anything until it has been battle tested in the fires of production hell.
The most common platform for high capacity work in anime studios is Windows NT 4.0 (Japanese). It can be administered by the staff (lots of reference books and users help), networks very well, has a large variety of powerful graphics applications and it's quite stable for the uses an animation company puts it to. NT has consistently come out the best in all our real-world tests and is (so far) the system that has proven to work best for us.
Macintosh systems are also in use at companies who started out with RETAS and for designers and illustrators. The Mac is seen as the "artist's computer" in Japan so many people getting into digital art work buy Macs and there is a pretty good knowledge base available. Japan has the most active Mac market in the world. Standalone system performance is a bit better than many PCs but we've found that there is a lot of performance degradation when the systems are networked, especially when they are put under stress.
A couple anime companies have Silicon Graphics systems which are used primarily for 3D. Although the price has come down somewhat they are still at least 50% more expensive than in the US and a network of 5 machines with applications costs more than many OVAs so it's not exactly cost effective. Many games companies (who tend to have much larger budgets than anime productions) use SGI systems though. (Never try to teach UNIX to artists. It is a painful and frustrating experience. I've had to have all references of UNIX purged from my memory so that I can direct creatively again.)
Most anime companies work with anywhere from 2 to 30 systems and many do not have a dedicated system administrator. (Most sysadmin people in Japan make 2-6 times what a reasonably well-paid anime company employee does.) Companies that have Internet servers and games departments usually have some UNIX systems for their main company servers and such but companies that size usually have programmers and can afford dedicated support people.
Most studios are constantly on the lookout for new tools that will help them create or manipulate images better. There are constantly new applications and upgraded versions of existing applications coming out so it's a full time job to just keep up with what's going on.