"Polyamory is a neologism, signifying having more than one long term sexual loving relationship at the same time, with the full knowledge and consent of all partners involved. Persons who enter into or consider themselves emotionally suited to such relationships may define themselves as polyamorous, often abbreviated to poly. The term is sometimes extended to refer to similar committed familial relationships that are not sexual in nature." [From Wikipedia]
Wikipedia has an excellent discussion of some of the issues this raises. Still it doesn't address the more practical points. Which partner do you go home to in the evening? What happens if two of your partners share the same birthday? Whose parents do you visit at Christmas?
All the normal problems of a monogamous relationship are suddenly mulitplied two, three or four-fold. I have the greatest respect for anyone who can even aspire to the ideals of polyamory, let alone pull it off.