To begin, let's strip all the humor from the joke by explaining it. In short, mixed dancing is not tznius. It leads to impure thoughts, touching (cross-reference shomer negiah), and who knows what! In fact, it's so untznius that some people believe it must automatically lead to inappropriate "relations" between the sexes and thus...babies. (Note: it's a difficult concept for some people, but these children are not necessarily mamzerim. Another subject for another day. But be glad you're just a convert and not a mamzer. Now that's a difficult shidduch situation.)
[As a sidenote for my own future shidduch prospects, I do not mixed dance. Or dance at all, which could lead to mixed dancing. Don't you dare assault my tzniut and imply otherwise.]
The most common use of "mixed dancing" as a joke is when someone names something that could lead to interaction between the sexes. Then someone says that "could lead to mixed dancing," and everyone laughs knowingly about all those crazy "relations" and babies that will result. Let's discuss some examples (and I apologize if they're awful because it's 4am and I just got home from the Grogger's CD release party, completely forgetting that the post I've been working on for tomorrow still isn't complete and isn't going to get complete at this hour).
I sat around Twitter for only 10 minutes before finding a good set-up for the joke. This joke could exist in a vacuum in space.
Unsuspecting Twitterer: "Shopping cart bumper cars. Holy carp"
Response: "Could lead to mixed dancing."
From a Doctor Who backstage clip...
Karen Gillan: [Talking about why she should never have a Twitter account] "I just don't...  I mean, one thing might slip, and you know...it could all..."
Internal monologue: Lead to mixed dancing. Zing!
And the winner for best use ever comes from a comment on a funny Jewlicious post:
[T]he beauty of this punchline is that it applies to all situations. Kind of like manna was supposed to taste like anything you wanted it to. Or maybe that was tofu. Still, the joke I heard was: “Why is premarital sex banned? Because it could lead to mixed dancing…”First-runner up goes to a comment on the Facebook debate about the probably-imminent mixed dancing at the then-upcoming Grogger's concert:
And if there ISN'T mixed dancing, I'm leaving. I'm sick of relying on brushing up against people on the subway for some action.
Other situations that lead to mixed dancing: kissing, television, college, Facebook, cars (especially women wearing seatbelts, which accentuates their upper torso!), dates, restaurants, and...uhh...everything. You're all doomed.
Quite frankly, with Jewish demographics the way they are, you'd think there'd be celebrating over every baby born, even if it was because of mixed dancing at a simcha. Hopefully one baby results from that wedding day, why not 30 more?
But in all seriousness, there has been a serious shift over the last 10 or 20 years (probably more) to separate the sexes in every possible situation. (Extra credit reading: The Great Mixed Dancing Controversy of 1960-61.) Some people say this is the true cause of the shidduch crisis: orthodoxy has removed all the "normal" situations when people might naturally meet, and singles are thus forced to rely on shadchans. This includes mixed seating/dancing/mingling at the meal following a wedding, social events, and even Shabbos tables! Then it gets translated to subways, buses, and anywhere you might accidentally brush against a person of the opposite sex. If you think this is funny, you should google the segregated seating buses in Israel (and a couple in the New York area). Some complete with a mechitza, from what I hear. You'll stop laughing.
Some have gone so far as to not allow "mixed seating" at dinners, even within their home. For instance, a family may host only single girls (and married couples) one Shabbat, but only male singles (and couples) the next week. As a single in a singles community, I have already heard the stories and seen the results of how many couples can meet at the Shabbos table, and I fully support it as a matchmaking tool. Further, a friend had great advice this week that a dating couple should never move beyond the starting stage of a relationship without sharing a Shabbos table. That way, you see how the other reacts socially - and you also get a kind of verification of his or her level of observance and Jewish knowledge.
Now I'm going to get some sleep. ...Which most certainly leads to imaginary mixed dancing. Opa!