Jack,
Another use of the term "aikikai" is loosely translated to "aiki association". Hence, many dojos have the term Aikikai as part of their name, ie, New York Aikikai, San Diego Aikikai, etc. And it simply means a place where aikido is done or headquartered.
As Rachel pointed out, more recently, "aikikai" not as part of a dojo name, has started to be associated with a particular "style" of aikido as practiced by Hombu dojo and such noteable shihan as Yamada Sensei (NY Aikikai), Kanai Sensei (New England Aikikai) and Chiba Sensei (San Diego Aikikai). There are other "styles" including Iwama, yoshinkai, etc, as previously mentioned.
Good luck with your training and visit as many dojos as you can. That's the best way to know what the right fit for you is going to be.
Best Regards,
Jack
P.S. Nice to have another Jack in the aikido community.