Wow, that's a BIG question.
First off, you're looking for Soo Bahk Do, Moo Duk Kwan as the generally accepted spelling. Formerly known as Tang Soo Do Moo Duk Kwan. The art was founded in 1945 by Grand Master Hwang Kee. It had gone through a few name changes before that (Hwa Soo Do, Kong Soo Do), but ended up on Moo Duk Kwan. The Tang Soo Do part is really just a generic name, it is the characters Karate translated into Korean. Following the Japanese Occupation, Moo Duk Kwan was one of the 5 major kwan that re-emerged.
When General Choi unified the Kwans, Moo Duk Kwan was one of the them that did not join and chose to stay a separate art. Hwang Kee considered TSD to be more traditional and did not want to move into the sport aspect of thing. Of course his reasons were much more deep than that, and no one knows the whole story, but he stayed separate. Over the years, the US Soo Bahk Do Federation has gone back and forth between the name SBD and TSD a few times. They have settled on SBD now due to the fracturing of the style.
Basically, when Hwang Kee died, his son took over, Kwan Jang Nim H.C. Hwang. Many of the older ranking Koreans split off from the Federation at that time, many before then as well. When they left, they took the name Tang Soo Do and Soo Bahk Do remained the "mother style," which is governed by the US SBD Federation. Today there are countless TSD organizations, but only one SBD organization. The World organization is known as the World Moo Duk Kwan and it excompasses all Soo Bahk Do outside the states. Basically, Moo Duk Kwan is the art, Soo Bahk Do is the style.
As for the style itself and what it is. Like most TMAs, it is composed of Hyung, Hosinsul, Illsoosik, kibon dongja, and daeryun. Many of the hyung were imported from Japanese and Chinese styles, but take on their own TSD/SBD flavor. The Hosinsul differs between TSD organizations, but within the US SBD Fed, it is completely standardized, so there is one set for the US. Same with illsoosik, one set for everyone. Basic movements (kibon dongja) look very Japanes in nature, but there are some distinctively chinese influences due to Hwang Kee's training in China. The sparring has many kicks, but probably more hand useage than Taekwondo. There are definately high kicks, but I don't think that there are as many "flashy" kicks as TKD - or at least not as many are used in sparring.
This is a VERY basic history and overview. If you want to know more, the website is
http://www.soobahkdo.com/fed-web/main-page.htm You can also look in the Tang Soo Do forum on Martial Talk. Though Soo Bahk Do and Tang Soo Do are no longer the same style, they are still basically the same, so the things you find in there probably apply. When you say Soo Bahk Do, though, you are generally referring to the one organization.