As Christ states the Aikikai is an organisation. It is probably the largest aikido organisation that covers many many different dojos in Japan as well as in the rest of the world. As far as I know they also handle hombu dojo - which is the 'main dojo' in Tokyo Japan.
Apart from running that dojo they also handle the international registration of all yudansha (black belt) grades.
Other organizations have been created over the years so that aikido can be found with little or no connection to the Aikikai. The reasons for this are numerous. Some want to emphazise differently in technique, some don't like the favourism that is bound to happen in an organization - especially one that is international...some just don't like organisations too much, and some think they are not given gradings as fast as they think they should. These are just some of the reasons.
All in all the Aikikai is the center of the cobb web that binds us all together. They supply a framework for organizing aikido, and they can lend credibility to any national organisation that meets the requirements. Sometime they even support us if we are starting something new or are in a difficult transition from one sensei to another (usually when the oldest one die or retire).
It is not perfect, but I like it since it provides a pillar on which to build our aikido. When Saito Sensei and NIshio sensei died in the early 2000's the two major Danish organisations that used to be connected to Japan through these two shihans had to find a way to work together and unify the contact to the Aikikai. Developing a new national organisation where we combine our forces and have one contact interface to the Aikikai has proven a good opportunity for learning new things and joining forces. This weekend we had a workshop where we all trained together at 20 different workshops on five different mat areas throughout a great saturday. Oh... and lot's of beer and good food afterwards
In my opinion Aikikai is like democracy. It may not be perfect but it is the best we have right now, and if one participate it is possible to help shape the organization and make it even better.
It is true that the value of the Aikikai has been debated a bit here lately. Don't pay too much attention to this. Right now you should just focus on having a great time learning a great budo.