AP : Thank you for your kind words ☺. Indeed, phaleristics, as uniformology, can be quite complex but I’ll try a short answer.
Student of History : The sashes you are referring to are parts of the highest class insignias of most Orders of Chivalry (Knighthood) or Orders of Merit. The badge (or cross) of these Orders are attached to these sashes. For some orders the sash is worn over the left shoulder, for others, over the right shoulder. There’s no special meaning regarding which shoulder they are worn on. The dark blue sash worn by modern (British) royals you mentioned is the « riband » of the highest British Order : the Order of the Garter (which has only one class).
http://en.wikipedia.org/wiki/Order_of_the_GarterAs shown on Serov’s portrait, Nicholas II also wore it :
http://www.flickr.com/photos/shnuggy/3269028715/The badge of the Order can be seen below his right arm, the star of the Order is shown on his left breast. He was invested into the Order (as Knight No. 802) by Queen Victoria in 1893 before he became Emperor.
The Imperial Russian Order of St. Andrew also had only one class. On special occasions, the badge was not worn suspended to the sash but rather from the Collar of the Order. When the collar was worn, the sash wasn’t, although a sash of a lesser order could be worn instead.
Other Orders had more than one class. Crosses (badges) of 2nd, 3rd or 4th class were not worn suspended from a sash but from a narrower ribbon either from the neck or on the breast, depending on the class.
More information about Orders of Chivalry here :
http://www.almanachdechivalry.com/index.htmlImperial Russian Orders :
http://www.almanachdechivalry.com/id9.htmlPlease note : long ago, in most armies, sashes were also worn over a shoulder as a rank insignia (usually for commissioned officers). In most cases, starting in the 18th or 19th century, they began to be worn at the waist (over the belt). So in his Preobrazhensky Regiment uniform, Nicholas II as 2 sashes : the sky blue one of a Knight of the Order of St. Andrew, and the silver one (with black & orange specks woven into it) common to all Russian officers (...except the Hussars which had another pattern). Still today though, for parades, some armies have sashes worn over a shoulder (e.g. British Infantry officers have crimson silk sashes, NCOs: scarlet wool). These have nothing to do with Orders of Chivalry.
Hope it helps!