A metamour is the lover of your lover. If you are also involved in a sexual or romantic relationship with your lover's lover, however, he or she is no longer a metamour and would be a paramour (or sweetie, or whatever alternative term you prefer to use, since "paramour" has unfortunately become loaded with negative connotations). The idea here is that metamour is someone you share a lover with but are not involved with yourself (except, ideally, as platonic friends or friendly acquaintances).
Contrast "metamour" with the concept of "the other woman" or "the other man" and with terms like "homewrecker". In the world of traditional American marriages and dating culture, there's not much room to view the lover of one's lover/spouse in anything but a negative light that casts shadows of drama, secrecy, lies, jealousy, betrayal, and competition over the whole relationship. In a monogamous environment (and the worlds of most romance novels) the other lover is seen as a poisonous serpent who must be stomped in romantic battle.
In polyamory, though, metamours are an expected part of the romantic landscape. The idea is that your metamour isn't a competitor who is plotting to take your lover away from you any more than your lover's other friends are set on taking him or her away from you. You both care about the same person, and you both know that bringing unnecessary drama into your mutually beloved's life is annoying and counter-productive. So, there's the sense that in a workable polyamorous relationship structure, all the people involved should be rational adults who will try to cultivate a good romantic ecosystem. Jealousy happens, sure, as do personality conflicts; the best thing to do is talk them out. But metamours should at least try to get along with each other rather than engaging in territorial posturing or, worse, making attempts at relationship sabotage.
An ethical question that gets discussed in conversations about polyamory is: "What consideration do I owe my metamours? If my lover is doing something I suspect will be hurtful to his/her other lover, should I speak up?" The answer varies from situation to situation. There's a fine line between friendly concern and being intrusive; your sweetie probably won't react well if he or she feels you're trying to control his or her other relationships. But, in general, if you realize that your lover is unwittingly doing something that his/her other lover will object to, it's best to give a gentle heads-up: "Hey, didn't you say you'd call Joyce at 10pm to let her know when you'd be home?" or "It's great that we've spent so much time together lately, but Joe might be feeling lonely ... maybe you should call and see if he wants to go out with you this weekend?" or "Are you sure that Amy doesn't mind that you're not buying her a birthday present?"
The shared lover has a critical role in all this. If you are in a poly relationship and decide to bring a new secondary lover into the mix, it's best to try to make sure that your lovers will be able to get along with minimal effort on their parts. Avoid dating irrational, possessive or emotionally unstable people (even if they're really, really hot). Further, it's a good idea to avoid dating someone you suspect your existing sweeties will find annoying or aggravating. Dating someone whom your existing lover knows and already dislikes shows a basic lack of consideration for his or her feelings, so it's best not to do it unless you're so smitten that you just can't help yourself (in which case, be prepared for a lot of compromises in other areas to make things work if you wish to keep both lovers long-term). If both lovers are local and share the same social circles, having to keep them separated at all times lest an argument break out is tiresome and not a lot of fun for anyone. I know a woman who decided to start dating a woman who set her husband's teeth on edge even when they simply shared the same gaming group. The subsequent friction between the two metamours led to truly breath-taking drama; I'm pretty sure they could see the smoke in Canada.