Hi!
Jujutsu is a term that came into colloquial use in Japan during the Edo period. In the following Meiji era, it spread outside of Japan and sometimes the term even "stuck" in place of what was really Kodokan Judo. So what we may call "Jujutsu" is technically not "jujutsu" at all. Confusing? Sound like a gray area? Yes it is confusing and a gray area and it's all in the name that the ryu gave it.
Now, that being said, there are more words used to describe "jujutsu" in Japanese martial arts than there are differences between them. That is to say, you'll find terms like taijutsu (BTW, Taihojutsu is modern and is seperate from TAIjutsu), kumi uchi, yawara, kempo (believe it or not), koshi no mawari and judo (an older word, later borrowed for Kodokan Judo) within many Japanese martial arts. These terms are all talking about what we in the West would call Jujutsu. Each school or "ryu" have named their "unarmed" techniques in a way that they felt was appropriate, so you have various names. That's that.
Joseph, depending on your location you may not be able to find "Japanese Jujutsu" so you may want to consider Judo for starters. It will introduce you to many solid principles that you may find useful in the future. Danzan Ryu Jujutsu enjoys a wide coverage in the United States as well as Small Circle Jujutsu. You may want to look around and see if you can find a place near you that teaches them.
Always,
-Russ