The dance movie begins with the Hollywood musical. Here, the dancing is often a way of bringing people together through a shared project (e.g., having to learn to dance) and, as such, it provides an impetus for a romantic relationship.
In these case, the dancing often ends up implying intimacy or sex, as here in the context of two people who dance together professionally but also have an offstage relationship:
Fred Astaire's sure they can work it out in Easter Parade
I actually think the whole of Easter Parade is a brilliantly well-observed examination of finding new love after a break-up, as seen through dance styles. It's significant, for example, that Judy Garland (New Girlfriend) has her own style of dancing and singing that is worlds away from the refined Ann Miller (Old Girlfriend). Fred Astaire (Heartbroken) has to accept and grow to love that difference before he can truly move on, and his own dancing develops and changes in the process.
Judy Garland awkwardly makes you remember your ex
The basic plotline of Easter Parade is similar to that of many other films of this era: boy loses dancing partner girlfriend, boy goes on Pygmalion project with new girl, boy falls in love with new girl, something goes temporarily wrong, but they all live happily ever after.
It's also, minus the dancing, the plot of Ten Things I Hate About You and She's All That, where the Prom replaces the Ziegfeld Follies as The Place To Prove Yourself (which tells us all sorts of things in itself).
It's partly for this reason that I keep asking people whether they think the dancing in these golden-era Hollywood films is diegetic or extra-diegetic (that is, whether or not we're supposed to assume the dancing is actually, literally happening in the story).
Obviously some of it is - when characters are on stage, for example, or reliving past thespian triumphs - but personally I think there's an important shift between this:
Yay, we solved a problem! Let's dance!
Your new dance partner is hotter than me
Whereas in the first clip (Singin' in the Rain) the dancing is a set piece, the action could happen without it, and if the characters were actually dancing as much as that in their day to day lives... well, that'd be a bit weird.
In Strictly Ballroom (whence the second clip), the dance moves the plot along, it's clearly observed and commented on by the other characters, and it fits within the context of the storyline (Scott and Fran have been practising for a dance competition). Significantly, too, it is interrupted, rather than brought to a show-stopping conclusion before next steps.
What about the singing?
Of course an important change is the loss of the singing, which in the earlier films acts as the extra step taking the action from quasi-diegetic to outright bizarre-by-the-standards-of-everyday behaviour. In fact, there's this fairly clear cut-off point, where dance films lose the singing accompaniment and singing films become instead consciously artificial camp-fests, where, in a nod to the more operatic tradition of musical theatre, the music - not the dancing - is clearly the star turn.
This category of films gives us works such as Moulin Rouge, Tim Burton's Sweeney Todd and Disney's Enchanted - which, by pointedly putting the singing back, jibes good-naturedly at the ridiculousness of the Disney tradition (its own) from which it draws.
But the effect of this separation is ultimately to free dancing up to go the other way - to become more naturalistic. With that, it also becomes more central to the plot of the film in question, and can even start acting as a transformative agent.
Meanwhile, though there's no full-blown singing from the characters, the music that accompanies these later dances is almost always vocal, and usually (as in that Strictly Ballroom clip) the song chosen comments on the action, Chorus-like, whether it's diegetic or extra-diegetic: so it's not so much that the singing has vanished, more that it's been caught and put in a stereo.
Dancing as socio-political project
Nonetheless, the dance films of the second half of the 20th century - from Saturday Night Fever onwards, say - generally begin to explicitly acknowledge the dancing as part of the plot. Usually, this means dancing becomes a catalyst for some kind of socio-political transformation - or, at the very least, makes some socio-political point (since famously - and bleakly - precisely nothing changes for the immigrant community out to forget in Saturday Night Fever).
This social impetus ranges from the upward mobility of the working-class protagonist in Flashdance - and its British equivalent, Billy Elliot, where the Power of Dance is so straightforwardly transformative the film was actually part-funded by Arts Council England - past the racial bridge-building of Save the Last Dance to the young, edgy sexual revolution represented by the eponymous moves of Dirty Dancing (which is contrasted to the middle-class, middle-aged dances of the rest of Baby's milieu).
I carried a watermelon
As Dirty Dancing made so abundantly clear, by this point the dancing was not so much implied sex, as a straightforward sexual statement - The Yooth having their day:
A little bit of after-work dancing
But building on its 1950s inheritance, late twentieth-century dancing can also be the means of personal growth (usually growth into a more challenging relationship), as in Strictly Ballroom and Shall We Dance?
Julia Stiles is confused
These traditions can also be turned to ironic, or even flat-out barbed, use, as in Napoleon Dynamite, Romy and Michelle's High School Reunion or the bleak, brilliant Fish Tank. Here, the dance set-piece (within a non-dance film) plays on the audience's understanding of the dance movie convention, and usurps it: there's the vague promise of redemption through a dance audition, but life's not like that. The closest you get is this moment of guarded family togetherness right at the end: