This is the name given to the art the Founder was teaching early in his development. It is very close in style to previously existing Jutsu forms such as Daito-ryu Aiki-jujutsu. It is considered to be one of the harder forms of Aikido. Most of the early students of the Founder began during this period and much of the early practice overseas was in this style (e.g. Abbe-sensei's teaching in the UK in the 50s).
Founded by Noriaki Inoue. Inoue-sensei was the nephew of the Founder and was originally associated with the early Ueshiba dojo. He has claimed that the art is different to Aikido but others have commented that it seems very similar to the Aikido of the early period.
This form was developed by Minoru Mochizuki, who was an early student of the Founder and also of Kano sensei at the Kodokan.This style includes elements of Aiki-Budo together with aspects of Karate, Judo and other arts.
For more see International Yoseikan Budo Federation website.
This is the style founded by Gozo Shioda. Shioda-sensei studied with the Founder from the mid-30s. After the war, he was invited to begin teaching and formed the organization known as the Yoshinkan. Unlike many later organizations, the Yoshinkan has always maintained friendly relations with the Aikikai both during and after the Founder's life.The Yoshinkan is a harder style of Aikido, generally concerned with practical efficiency and physically robust techniques. It is taught to many branches of the Japanese Police. The international organization associated with the Yoshinkan style of Aikido is known as the Yoshinkai, and has active branches in many parts of the world.
For more see Aikido Yoshinkan website.
In recent years, there have been a number of offshoots of this style, usually developing for political reasons. These include the Yoshiokai, headed by Takashi Kushida-sensei.