Seitei Iaido is the Zen Nippon Kendo Renmei’s (全日本剣道連盟) Iaido kata set which was put together from various koryu kata. the Seitei kata are the standard all over the world for gradings and as a standardised form of Iaido. Traditional Iaido or Koryu was formed in the Edo jidai of Japan, over 500 years ago. The kata are all different depending on the Koryu. Each koryu was the creation of the founder of each style, usually some enlightened Samurai of the time. Each Koryu has its own philosophy which can often be seen in the higher levels of their kata.
Seitei is a form of Iaido which is the same wherever you train, whereas Koryu changes from style to style and sometimes from place to place as well. Seitei iaido was created by the ZNKR after WWII as a way to study kenjutsu in a peaceful context. Directly after the war, the Japanese were forbidden to own weapons and to practice martial arts. As a result, Iaido was re-worked to make it very clear that practice in the use of the sword was for a Zen purpose rather than a martial purpose, and various Koryu kata were chosen and slightly changed and standardised, and hence the ZNKR Seitei iaido came about. In Japan (and hence the world) Iaido is organsied either by the Zen Nippon Kendo Renmei [ZNKR], or by each Ryuha's Soke which is the head of the style who will oversee their particular style. Gradings in Iaido only have legitimacy if they are given by the Soke of the Ryuha or the overseeing body.
The difficulty with koryu rankings is that they are not really standard. If you can say you are 5th dan of some koryu ryuha, it is hard to equate that to a set standard. With Seiei, the rankings come from the ZNKR which is standardised and recognised all over the world. If you can say that you have 5th dan from ZNKR, everyone has a good idea of what that means Seitei iaido is usually practised first in most ZNKR affiliated dojos because it is generally more neutral than Koryu Iai. Koryu Iai also has a premium status of being the actual style created by the founder hundreds of years ago. During these times, and often still now, techniques and philosophies of each Koryu are a closely guarded secret. Most Koryu have naiden or okuden levels taught only to the highest level students, while Seitei iaido is completely open and there are no secrets.
Doing Seitei is a good way to rid yourself of bad habits in your Iai. Seitei requires you to be straight and exact in your execution of the kata, whereas Koryu kata are often more flowing and seem more natural. Doing Seitei first allows you to have sufficient control over your body and sword to be able to not fall into bad habits. Straight, centered and correctly aligned are concepts that are important in Koryu too, but Seitei makes it really clear by picking exact lines of centre and horizontal positions. Even though Seitei is not hundreds of years old with a direct samurai lineage, it kind of is. The seitei kata were created by the guardians of this knowledge, the people who have the best claim to Iaido legitimacy in this modern world, and are based on the old kata handed down to us. Combined with the clear emphasis on reigi, seishin-shugyo and non-violence, this makes Seitei Iaido as at least a legitimate Budo as Koryu Iaido.
Seitei iaido is updated and changed regularly to keen up with what is decided to be the best interpretation of Iaido as a Budo. Old kata are reviewed and new kata are added periodically so it is always a challenge to keep up and you must always strive to be better. The video's below show you what Seitei iaido is like. The details of Seitei do change from time to time, so no video will be accurate for very long however. These videos are a little old, but will give you the general idea.