These definitions are our own, but we've found them useful in describing polygamy...

• FLDS, the Fundamentalist Church of Jesus Christ of Latter Day Saints, commonly known as the Fundamentalist Mormon religion, which embraces plural marriage as a sacred principle.
   
• LDS, the Church of Jesus Christ of Latter Day Saints, commonly known as the Mormon religion, which abandoned plural marriage as a sacred principle in 1890.
   
• Group marriage, a communal marriage of more than one husband with more than one wife.  Group marriage is extremely rare, outside of science fiction, and has little to do with polygamy.
   
• Menage a trois, from the French, literally, a household of three (lovers).  Some menages a trois are polygamous.
   
• Plural marriage, synonymous with polygamy.
   
• Polygamy, a family of one husband, two or more wives, and their children.  Technically, this is polygyny, as opposed to families with one wife and two or more husbands, which is called polyandry.  Such families are rare, and polygamy has become the common name for polygyny.
   
• Polyamory, a network of open and closed sexual relationships. Although polyamory could include polygamy, most polyamorous relationships are not polygamous.  The focus of polyamory is almost always on sexuality.  The focus of polygamy is always on family.

Further glossary items can be found at Jay's Thoughts on Stuff.