Polyamory (or multiamory/polyphilia if you dislike the combination of Greek with Latin, but the word already exists, is already in common usage and you just really need to deal with it, polyglots) is the opposite of monogamy, basically. It's "the philosophy or state of being in love or romantically involved with more than one person at the same time."
Now, the first reaction of most people who've been raised in a monogamous society is to reject the idea as selfish, immature, unhealthy, etc. In actuality, selfishness and immaturity are counterproductive to a polyamorous lifestyle. They have no place in any healthy relationship -- and monogamous relationships are just as prone to the exact same issues. As for healthy? Any relationship can be healthy or unhealthy depending on maintenance, maturity, commitment, consent, etc. Read more about myths of polyamory.
I once didn't 'get' polyamory. I couldn't understand how you could love more than one person at a time. Then I realized how silly I was being -- of course you can! Any mother of more than one child already does. You don't stop loving your firstborn for your second, or your third, etc. Your love isn't a cup to be filled, but a bag of holding without any weight limit.
Not everyone is built to do this with romantic relationships. There are people who want to say that poly is the 'natural' state, but I feel that's a slippery slope. It falls into the narrow minded idea of 'normal.' The fact is that there are multiple states of 'natural.'
Not all humans are naturally poly, just as not are all naturally mono or even sexual. They're all variations of 'normal.' (quick rant to poly people who pull this card: yes, one person can be everything you need in one type of a relationship, and be perfect for you. Don't go anti-mono just because you're not. That's as bad as being anti-poly.) To say otherwise is like saying everyone is naturally bisexual or everyone is heterosexual/homosexual, etc. It's just not true.
Now, to explain the different types of poly relationships:
An open relationship is when you are actively allowed to add to your existing relationship(s). A poly relationship can be closed just like a monogamous one. This is called polyfidelity. This can change according to the needs of the relationship(s)/people in them. One thing you must be in poly relationships is open and communicative. If your communication is poor, it will fail (or at least struggle until everyone is miserable).
"Swinging" is purely sexual. It's a relationship that is open to trade/swap partners or explore outside sexual relationships without attachment. Swinging can be a risk to the couple at the core if attachment occurs, as this usually occurs in couples that are not willing to share emotional relationships (and note that they are still a 'couple' -- this is a two person bond that explores outward typically). Read more on swingers.
All dynamics are different. Some groups choose to include marriage, some do not. There is no 'right' or 'wrong' as long as healthy communication and full consent are involved.
Some people use a hierarchy system while others feel that's demeaning (and how the hierarchy is established varies as well -- some by 'importance,' some love vs sex, for instance, and some by time spent).
Some are mono-poly practicing, where one member is monogamous and the other is polyamorous and the mono member is okay with the poly dating. Some are mono-poly closed, where the poly member has agreed to live monogamously to respect the wishes/belief/heart of his/her significant other.
Mono-poly relationships are tricky in that most people are jealous, but especially when one cannot understand the drive of the other. Selfishness of any kind can easily destroy the mono-poly relationship. Feeling that you aren't being allowed to explore your sexuality/ability to love on the poly's part (this person may have regular trouble in poly relationships as well, because selfishness can lead to jealousy as quickly as possessiveness does, so therapy would be in order regardless of their decision to remain mono-poly or leave to become full poly) or being unwilling to accept that the other person has different emotional needs on either part or feeling that your partner doesn't 'love you enough' because they can't prevent themselves from falling in love with someone else.
Poly people strive for an emotion currently termed 'compersion' -- this is finding joy in the happiness of those you love, unselfishly. I actually strive for that myself. I'm mostly there, but when my needs aren't being met, I lose it (which is completely normal).
Poly-poly relationships come with social stigma and are more work and maintenance. All relationships require work and maintenance. Poly people who are committed to healthy relationships are just as equipped to deal with the extra work as mono people.
Small note: Helper personalities (people who want to fix things, mediate -- people like me) can be the utter death of a relationship if they can't learn to shut up and let the others figure it out on their own. They either stress themselves out not being able to help or accidentally sabotage other people's relationships because they are trying to push for communication that the others aren't ready for. Therapists are both great and terrible in poly relationships.
Both monogamous and polyamorous relationships have the potential for amazing rewards. There is no superior/inferior here. Only what your romantic orientation leads you to.
