I am still new to the Bujinkan with just under a year of training, but I am very curious of how other dojos present and train the Godai...
Godai (五大) : "Five Greats". A belief system from Tendai sect Buddhism using five elements as a fundamental structure to describe relationships between the physical reality and "emotional" reality. (Tendai sect Buddhism is an esoteric form of Buddhism found in Japan).
Gogyo (五行

: "Five Lines", "Five Ways" et al. Originally a system of describing the physical world developed by Chinese Taoists. Later imported to Japan through Feng Shui and adapted as an ordering or numbering system.
Gogyo no Kata (五行之型

: In the Bujinkan, the Gogyo no kata was brought into the Tenchijin ryaku no Maki from Gyokko ryu by Hatsumi sensei. Basicly, its just five fundamental kata to teach smooth movement, proper striking, etc. There is no relation to an "elemental feeling" for each drill.
Godai no kata: As far as I know, the Godai kata were developed by Stephen Hayes, and are taught within his Toshindo system. The physical movements came from kata Stephen Hayes learned from Hatsumi sensei.
The movements of the Gogyo no Kata and the Godai are COMPLETELY DIFFERENT. The purpose behind the motions themselves are different as well. In the past year I've had quite a few people ask me what the difference between the Gogyo no Kata and the Godai are; I think because the names sound so similar, and the fact that thet both use "Earth, Water, Fire, Wind, Space" (even though they are used in different ways)there has been some confusion.