Polyamory
Polyamory is the expansion of romantic relationships beyond two people with the consent and knowledge of all parties involved.
It can also be used to describe an orientation or your desired relationship structure.
Its use in two contexts is similar to how monogamy is used.
"We're in a polyamorous relations" describes the structure of relationships.
"I'm polyamorous" describes your orientation towards the type of relationship you seek.
Polyamory is often shortened to "poly" or "polyam".
When used as a reference to relationship structures, polyamory is a subset of (or ethical non-monogamy).
These are some broad categories that are helpful to understand.
For a more complete picture of all the ways you can define your relationships, check out this graphic.
Source: The Map of Non-Monogamy
Polyamory doesn't factor in legal or religious marriage. Historically, Polygamy has been the term used for a marriage with more than two people. This term has fallen out of use due to its connection with unethical religious and cultural practices.
Polygyny is a marriage of one man to multiple women.
Polyandry is a marriage of one woman to many men.