This is the style of Aikido I practiced.
It was found by Prof Kenji Tomiki, a senior student of Morihei Usehiba (founder of Aikido) and Jirgo Kano (founder of Judo). I believe he was awarded the rank of 4th Dan in Judo and 8th Dan in Aikido.
Kenji Tomiki's study of Judo influenced his study of Aikido. While working at Waseda University in Japan, he tried to set up an Aikido club in the University. As the university only granted access to the gymnasium facilities to competitive sports, the university would not allow Tomiki to setup an Aikido club as it was not a sport. To get round this, Kenji Tomiki came up with a set of rules to govern a newly created of Aikido Randori. As such, the University allowed him to set up his club as his Aikido now had a competitive aspect to it.
Usehiba Sensei and the Honbu weren't too happy about this though. So Kenji Tomiki continued to refine his Aikido as he saw fit, developing the use of Aikido using randori.
So that is the basic history as I know it!!
In terms of the actual randori and competitions, there are two areas:
Tanto Randori (Uke has a Tanto, Tori is empty handed)
Toshu Randori (Both Uke and Tori are empty handed).
Uke is the "attacker", Tori is the "defender". Those are terms used in the English Language, though I am led to believe they don't properly define the Japanese words of Uke and Tori.
In Tanto Randori, Uke is essentially trying to stab Tori with a wooden/rubber tanto. Tori must stop Uke using "free" variations of Tomiki Kata. By "free" I mean that you can interrpt that kata in any which way you can imagine!
In Toshu Randori, both Uke and Tori try to put techniques (such as a wrist lock) on their opponent while their opponent resists and tries to counter the technique with a movement of their own (such as a throw). The best description I have heard for Toshu randori is "Japanese fencing without the sword".
Here's a decent example of Toshu Randori: http://www.youtube.com/watch?v=KRlKQpn1RXc
Here is some Tanto Randori, though the people here seem to be going at a pretty casual pace:http://www.youtube.com/watch?v=j-0bjAFgIZ8
In conclusion, I would recommend checking it out, it is an awesome system of Aikido!
Sorry for the long post! Anything that is not clear please feel free to ask!