Re: Translating "styles"
It's really hard to say how many styles there are. There are really as many 'styles' as there are teachers.
For me there are six major branches of "Aikido"- The Yoseikan branch, the Yoshinkan branch, the Tomiki/Shodokan branch, the Iwama branch, the Aikikai branch, and the Ki no Kenkyukai (Ki Society) branch.
These six seem to have the biggest influences and biggest distinctions of the Aikido family. But it gets really tricky, because there are lot's of great students of the founders that created their own thing, and I'm not really being fair by leaving them out.
But for me these six represent most of what we call Aikido today. Of those, the Aikikai and Yoshinkan are the biggest, and represent a lot of what you see in the Aikido world. However depending on where you are other branches have a good foothold. I used to only see the differences in Aikido 'styles', but as I've come to understand Aikido better, they all share more then they differ.