There's always more that can be said, but I have a previous answer to share (please note that the original question was about a slightly broader period of time):

The tl;dr is: yes and no. I'd also note coming in that the phenomenon did not stop at the eighteenth century, and that we can still see it in recent times with things like the Macarena, Cotton-Eyed Joe, the electric slide, etc.

Reconstructing historic dance forms is extremely difficult. You have the first difficulty in that it's hard to adequately explain how to dance through text, and then the added fact that people simply didn't bother to write down how to dance for the most part, as dance was and is generally taught in person by an experienced teacher.

However, there were dance manuals produced throughout this time (or at least after the development of the printing press, so from the fifteenth century on), whether to instruct a solitary individual or to be an aid to dancing masters. The purpose of dancing wasn't just to have fun, but to learn physical grace in everyday life, and to show off your knowledge and training in company, so such dancing masters were common! All dance amongst elites (who this entire answer is basically about) was a performance, even if you weren't on the stage: when you attended a ball, there were many opportunities for people to watch you, from your partner to other dancers to bystanders. All movement really was as well. And as you would expect with that context, these manuals often talk quite a bit about being poised and graceful rather than simply giving you the steps.

Position, then, is the different Placing or Setting our Feet on the Floor, whether in Conversation or Dancing; and those for Conversation, or when we stand in Company, are when the Weight rests as much on one Foot as the other, the Feet being considerably separated or open, the Knees straight, the Hands placed by the Side in a genteel Fall or natural Bend of the Wrists, and being in an agreeable Fashion or Shape about the Joint or Bend of the Hip, with the Head gracefully turning to the Right or Left, which compleats a most Heroic Posture; and, tho' it may be improper in the Presence of Superiors, among Familiars, it is a bold and graceful Attitude, called the Second Position: Or when the Heel of the right or left Foot is inclosed or placed, without Weight, before the Ancle of that Foot by which the Poise is supported, the Hands being put between the Folds or Flaps of the Coat, or Waiste-coat, if the Coat is unbuttoned, with a natural and easy Fall of the Arms from the Shoulders, this produces a very modest and agreeable Posture named the Third Position inclofed:

(From the first chapter of The Art of Dancing Explained by Reading and Figures: Whereby the Manner of Performing the Steps is Made Easy by a New and Familiar Method (1735))

Now, in the earlier centuries of this stuff, dance masters were frequently full-on choreographers: that is, they were there to teach courtiers and/or paid entertainers a set of dances for a performance - the sort of thing that would eventually come to be called a masque or ballet in France and England. By the late fifteenth century, Italian courts were featuring grand spectacles involving moving scenery, purpose-written music and poetry, dramatic scenes, and dancing between acts or as part of the drama. Members of the nobility and royalty, particularly female members, would show off their taste and money by commissioning, funding, and partaking in these performances. However, these choreographed pieces were typically just more elaborate and rehearsed versions of the dances that elite people were doing themselves. For instance, in the 1487 ballet given for Lucretia d'Este's wedding, the goddess Diana performed a slow bassedanse/bassadanza with her nymphs in the same way that wedding guests might themselves in a more casual setting or sometime after the show. Other early Early Modern dances include the galliard, the pavane, the branle, and the tourdion, which Susan di Guardiola has done good work in reconstructing based on the sixteenth century French dance manual, Orchesographie.

Dances performed together in groups as part of a ball or a performance continued to be common among the elite in the seventeenth century, although in England the older forms were being supplanted by the "country dance". A number of these were preserved in John Playford's English Dancing Master (1651), a book that's been enormously influential among dance reconstructionists. It consists of a list of tunes, each tune with fairly specific instructions for steps that give a direct picture of what was being danced. (A minor and pedantic clarification to "danced in groups", though. Playford's figures were frequently written for groups of three couples, sometimes specifying things for the first, last, or middle couple to do; the floor could, however, be filled with a number of six-person groups performing the dances together.) The instructions to go with the tune "The New Bo Peep", "longways for as many [couples] will", run:

Lead up all a [double] forward and back. That againe:

[Women] goe all to the wall and stand,

men go up to your owne [women]. and peepe four times

on each side behinde them, fall to your places all and turn single.

Sides all. That againe:

As before the men going first.

Armes all. That againe:

As the first time.

Playford's book was reprinted and updated many times between 1651 and 1728, after which time it was obsolete and old-fashioned, so should not be taken - as it frequently is - as the method of fashionable dancing by the late eighteenth century, let alone the early nineteenth. Other dance manuals were written, and other combinations of figures became fashionable. Now, these country dances break from the concept of "choreography" because despite the association of a set of steps with a tune in this or that dance manual, none of it was hard and fast. You might find the steps given for "A Trip to London" in one book with the tune "Lark in the Meadow" in another, and in any case, country dance figures were frequently set by the couple at the top of the line, rather than being statically attached to the music. The couple might ask for the song, "A Trip to London", and then start performing a generic and popular set of figures, essentially "teaching it" to the couples below them (who likely already knew them, or could easily pick them up) as they moved down the line in the dance. It's still choreographed in the sense that it's a fixed body of step-types that everyone is taught as a child, but it's not like "The Duke of Kent's Waltz" was struck up and everyone went, "Ah, TDoKW!" and performed that dance.

But in the early nineteenth century, this was becoming old fashioned. New dances branched off from this in two interesting ways. First, couple dances where two partners stayed together for the whole dance, never interacting with anyone else: the waltz was the first of these, developed on the Continent around the time of the Napoleonic Wars, and it was considered very lewd in England when it was brought there. Others would follow, like the polka and the schottische, and eventually couple dancing would be the standard form of ballroom dance. The other branch was a return to more set dances performed by groups (in opposition to the country dances performed in long lines of couples): first with the cotillions, which were supplanted by quadrilles, which would become square dancing by the end of the century. Quadrilles would also tend to be more choreographed in that they were set steps that went along with a title, but people didn't think of them as "choreographed", with that connotation of being the product of a director's vision - they thought of them as set dances to be learned.

Some sources and interesting links:

A massive collection of historical dance manuals

Women's Work: Making Dance in Europe Before 1800, edited by Lynn Matluck Brooks (2007)

"On the Question of Pictorial 'Evidence' for Fifteenth-Century Dance Technique", by Sharon Fermor, Dance Research: The Journal of the Society for Dance Research (1987)

Issues of Dance Notation: Domenico da Piacenza’s Dance Writing in XV Century Italy, a thesis by Chloe Spedding (2007)

To answer your question (before getting into the cultural background of these dances and how they differ from what you'll see in a film), these depictions seemed to be based in England, and during the 18th century, England's "dance scene" was heavily influenced by France. The sorts of dances that would have been en vogue in the 18th century would have been lingering on the edge of style by the time the 19th century rolled around. In the 18th century, dances like the bourée, gavotte (for music with two beats per measure), chaconne, minuet (for music with three beats per measure), canaries, gigue (for music with compound duple rhythm). There are 350 dances known from that period - with a basic set of 20 or so steps / figures and subtle variations used.

In the 18th century, there was an emphasis put on English country dances (dances originating in England) and later, the waltz, a dance that was considered exceptionally scandalous at first, and not widely danced in formal balls where the wealthy were hosted and expected more socially "correct" conduct. Country dances like the contradances or longways style dances that are most often seen in Regency era movies were made up of long lines of dancers performing many different figures, ranging in complexity - a "caller" would choose the dance and the tune. Again, there are many different dances, hundreds most certainly, some with rather creative names like "Money in Both Pockets", "Duke of York's Fancy", "the Balloon", etc. Also popular was the Reel (sets of fancy steps in place and interlacing), the Cotillion (a French dance, danced in a square) and later the Quadrill (shorter version of the Cotillion).

How people would learn these dances would often depend on their social standing. For the wealthy, employing a dancing master was an essential part of a young person's education. Even for the aspiring members of the middle class, having lessons would be advantageous for furthering their social ambitions. In Charles Dickens' novel "Bleak House", fictional character Caddy Jellyby - solidly in the middle to lower-middle class strata - decides to "better herself" by taking formal dancing lessons from a father and son who teach both dancing and deportment (though the father mostly lives off the son's income).

Accompanying lessons (or instead of lessons) there were numerous books written with dance figure notations, music and instructions - as well as volumes solely on dance etiquette. Here's an example of a dance notation, showing the music, figures and progressions: the Rigadoone

Now, the issue with many of the movies you'll see showing dances from the Regency era (early 1800's), is that these dances are far more stationary and lack the ballet like figures, jumps and other elements that would ruin that sense of restraint exhibited in the dances on the screen. They would have been more bouncy and lively overall and not lending themselves to this elegant sort of image movie makers have created.

At these balls there were usually a dozen dances in an evening; Jane Austen comments on dancing 9 of 12 dances at a ball in a letter she writes detailing the event and an article covering a Royal Ball in 1813 mentioned 9 dances in total. The Master of Ceremonies at a ball would ensure that etiquette was maintained - people could be expelled and fined for misconduct - he would also arrange introductions between parties. Clapping, snapping your fingers or "howling" at a country dance was discouraged. You must not leave a dance partway through - you must not call a dance that you cannot subsequently perform (you must call another). A couple not dancing when a dance is called by a lady shows great insult to her.

The waltz was considered a terribly scandalous dance at the time it arrived in England, called at times "the godless Spinner". The Prince Regent's Grand Ball featured a waltz in 1816, and the The Times of London reported on the occasion: "We remarked with pain that the foreign dance called the Waltz was introduced (we believe for the first time) at the English court on Friday Last.  …it is quite sufficient to cast one’s eyes on the voluptuous intertwining of the limbs and close compressor on the bodies in their dance, to see that it is indeed far removed from the modest reserve, which has hitherto been considered distinctive of English females.  So long as this obscene display was confined to prostitutes and adulteresses, we did not think it deserving of notice; but now that it is attempted to be forced on the respectable classes of society by the civil examples of their superiors, we feel it is a duty to warn every parent against exposing his daughter to so fatal a contagion.” You can never catch a break when you succumb to the latest dance craze. But we all know what dance dominated the dance floors in the 1800's... it was the "godless Spinner", now accepted by the masses. 😆