An "open" relationship is one in which one or (ideally) both partners are variably free to seek out sexual or romantic relationships with others. I've known a lot of people who've been involved in open relationships, mostly various types of polyamory rather than swinging. You probably know people in open relationships but just don't realize it. With the exception of a few particularly liberal urban areas, our culture in the U.S. is far more accepting of infidelity than it is of consensual nonmonogamy, so swingers and people in polyamorous relationships frequently won't tell anyone but very close, trusted friends.
Sometimes a couple will decide to open their relationship because they've realized neither of them is wired for monogamy. Other times, it's to address issues like wildly mismatched libidos, differing needs for emotional involvement, or because one partner is keenly interested in exploring BDSM and the other isn't. Sometimes, a relationship is opened for the sake of helping one or both partners explore their sexual orientation. Or it's opened because one partner is dedicated to a demanding career that involves long hours on the road or at the office. Specific reasons are as numerous as the people involved.
The decision to open a relationship is not to be made lightly. If you can't bring yourself to talk frankly with your partner about emotions, or sex, opening your relationship is probably a bad idea. If you feel you're being forced into opening the relationship, or if you're issuing ultimatums to your partner to allow you to see other people ... it's a bad idea; both people have to be genuinely willing to try the arrangement or it won't work. If a relationship has deep flaws, adding more people to it often causes those tiny cracks to fracture, hard, and the relationship fails. Sometimes, one partner will lobby for opening a relationship while he or she secretly wants to use the change as a testing ground for separation or divorce. But if both people are equally dedicated to each other and to making their relationship work, and they realize that they can't fully meet each others' relationship needs, opening a relationship can enable people to stay together.
Open relationships are seldom the sexual anarchy free-for-alls of constant orgies that some monogamous people might imagine. They mostly look like any other relationship ... just with more people involved. Sometimes, they are "don't ask, don't tell" type arrangements, but those seldom seem to work out well in the long term. From what I've seen, the people in successful open relationships do a lot of talking about anything and everything with their partners; secrecy and dishonesty are the opposites of openness and are the hallmarks of cheating.
From the standpoint of trying to maintain honest and open communication with your partner, it's important to discuss and mutually agree on ground rules when opening a relationship just so everybody knows what's expected. Good expectations management can help prevent a lot of arguments and hurt feelings down the road. Talking about and establishing rules can also help many people feel less anxious about losing their "place" in a relationship.
Many rules will be a matter of logistics ("Please be home by 6am so you can help me get the kids ready for school" or "I'd rather you didn't look for a third girlfriend; when would you have the time?") or health ("Please ask her to get tested for STDs, and I want you to always use condoms even if she comes back clean.") Some rules are negotiated for the sake of personal comfort ("I don't mind if you bring him to the house, just please don't fool around on my grandma's quilt.")
I've also seen emotional limits put on outside/secondary relationships. One restriction that is frequently uttered is, "I don't care if you have sex; just don't fall in love!"
This one might sound like a good idea ... but it's not.
First, your partner's love isn't a pie. Preventing someone else from having his or her love doesn't mean there's automatically more love for you. In fact, if the restriction makes your partner feel resentful or unhappy, over time there's likely to be a whole lot less love coming your way.
Second, it speaks to an emotional insecurity on the part of the demanding partner. If you trust and love your partner enough to allow him or her to share his/her body with others -- which potentially exposes you to incurable diseases like HIV, herpes, and HPV even if they religiously use condoms and dental dams -- you should feel secure enough to know that your partner cares about you and will come back to you when he or she is done visiting or playing. If you genuinely worry that your partner doesn't care about you and won't come back ... see my comment above about deeply flawed relationships cracking under the weight of other people. It will be painful, but it's better to end a relationship with an indifferent, irresponsible or neglectful partner than to go along with opening the relationship in the hopes of keeping him or her in your life.
It's okay to be scared. If you value your relationship, of course you want to hang on to your partner, and so of course sharing your partner will be scary.
But if you impose your own fear on your partner and try to unreasonably restrict him or her, it's not fair to your partner, and it's not fair to their partners.
And, yes, telling people that they're not allowed to feel specific emotions is completely unreasonable. It's always a losing proposition. Always. Telling someone not to fall in love with the people they share physical intimacy with is just as futile and unreasonable as telling someone they have to fall in love with the people they play with at a BDSM party. It's like saying "Hey don't get mad!" right before you sucker-punch someone in the stomach.
By telling partners they're not "allowed" a certain emotion, you're disrespecting their feelings. You're telling them to stunt and numb themselves emotionally. You're telling them to lie to themselves about what they're feeling at any given time ("Bob said I couldn't fall in love, so ... I'm not in love! I'm not!" sticks fingers in ears "La la la la ....") You're telling them to live in denial.
Do you seriously think that putting your lover in that kind of head space will be good for your relationship?
Love happens. Fear happens. Emotions happen. They're not a choice.
What a person does about his or her emotions, on the other hand? Totally a choice, and totally controllable (aside from physiological responses like crying jags; some days the tears just won't stop, and you can't help that, but you've usually got a choice about where you shed them).
Falling in love doesn't mean you're bound to break your promises or abandon the other people you care about any more than getting angry means you're bound to menace the target of your anger with a knife. (If, in fact, you have a hard time controlling your reactions to your emotions, and you are the sort to threaten people with weapons when you are enraged, I humbly suggest that for the sake of your prospective metamours you do not pursue an open relationship.)
If you decide to open your relationship, don't let your fears drive you to make unreasonable demands on your partner. Work on your emotional security first. Talk it out. Speak your mind and voice your concerns honestly; you have the right to expect your partner to listen to you and respect your emotions, but you also have to be prepared to do the same for him or her. You're both in this together.