Before music went electric, music fads were most often based on the dance step, rather than the style of music, because professional bands were usually expected to play in a variety of different styles. Whatever would attract an audience (which, before recorded music, mostly meant dancing), then that's what a professional band would play.

In the U.S., one out-of-fashion dance step in the 1920s was the quadrille. This was a dance that had been quite popular off and on during the 19th century in both Europe and the United States. (There's a chapter dedicated to the dance in the 1865 book Alice's Adventures in Wonderland, for example.) But the quadrille craze was well over by the 1920s, so when the Los Angeles Herald heard it was making a comeback in 1921, they wrote:

"They shimmy not, neither do they tottle. The debutante will soon be taking dancing lessons from grandmother. The quadrille is coming back."

A 1924 syndicated article about sixty-year-old Henry Ford made a similar joke, and added the "reel" as another example of an old-fashioned dance:

"Henry Ford was the liveliest of the crowd at the housewarming given his brother-in-law, M.D. Bryant, last night, dancing the reel, quadrille and all old timers."

An Indianapolis article from 1924 about a "big celebration" in Fresno, California, also called out the quadrille and the reel as being old-fashioned:

"A tent 180x250 will be used for old-fashioned dances featuring the quadrille and Virginia reel. Klansmen everywhere are invited to attend."

In addition to the quadrille and the reel, the march was another out-of-fashion music style of the 1920s. John Philip Sousa was still on tour with his band for most of the 1920s, and apparently attracted an older (albeit enthusiastic) clientele. In a review of a 1922 concert in Champaigne, Illinois, the reviewer noted that a Sousa concert was not complete until he'd played his most famous march, "Stars and Stripes Forever". But they also noted that Sousa's attempts to work contemporary sounds into his set weren't convincing. The addition of a "Lively Flapper" with "bobbed hair" who sang a song, and a xylophone solo later on did nothing to convince the reviewer -- they were "just made for jazz". The article went on:

"In some cases, we wondered if the band were not paying especial attention to the fact that a college audience was listening, on account of the slap-stick crashes, pistol shots, jazz and other noises used to get a 'kick' into the crowd."

Nevertheless, the reviewer ended with: "Once more the March King justified his title last night." Sousa "thrilled" the audience with all the "old favorites", and the audience responded with the "usual prolixity of encores".

That same year, in 1922, the trade magazine Music Trades predicted the "return of old-fashioned ballads". The article called the "foxtrot" the hip, new dance, while the "waltz" was a dance that was out of fashion:

"Despite the clamor for jazz and more jazz, the old-fashioned ballads with pure rhythmic melodies are fast coming into their own. Many of these are ballad fox trots which bring with them the grace and charms of the old fashioned waltz, which meet the summertime requirements with their very ease, rather better than the peppery, just-can't-keep-still jazz tunes."

If any more proof is needed, in the 1921 catalog for the major music label Victor, the company was emphasizing the hip dance steps the "foxtrot" and "one-step" on their recent releases, while both the waltz and the reel are suggested to be old-time music -- though they still had plenty of waltz records for sale.

TL;DR: The quadrille, the reel, the waltz, and the march were among the styles/dances that were seen as "old people" music in the 1920s. Jazz, and the "fox trot", were the music and dance crazes of the time.

There's always more to be said, but several things that /u/hillsonghoods answered very well to a similar question stay with me. In particular I didn't realize that the the concept of the American teenager didn't exist until a while (and a couple of eras) after the time period you reference.

tl;dr Old-time music

In the 1920s the growing radio and recording industries started to expand the types of music they offered and started to record a variety of regional and ethnic genres. One style of music that record companies sought out was old-time music (also called old-time or old-time country music), which was specifically marketing as old-fashioned or from a bygone era.

Now what is old-time music? As it's used today, old-time music is usually broadly used to refer to traditional string band music. For those who have never heard it it's often described as an ancestor of bluegrass and country music. It's usually associated with the fiddle and banjo but also includes other instruments like the dulcimer, harmonica, guitar, or even a capella or unaccompanied singing. It is sometimes called "mountain music" or "hillbilly" music because it is most associated with Appalachia, but it's been played throughout the South and urban cities as well as rural areas. Ruth Seeger called old-time music "a folk name for folk music". Her son Mike Seeger (brother of Pete Seeger) said "to me, southern old-time music is the home-grown, self-made, mostly rural music for people's everyday use learned and 'performed' person-to-person before the advent of the electronic age in the nineteen twenties". He touches on the key points that folklorists and ethnomusicologists use when talking about old-time or folk/traditional music in general: music that is passed down between people and generations and shared among a community or region. Other users have mentioned dance, and old-time music is closely tied to square dancing and before the rise of the recording industry it was usually performed at community dances or at home for entertainment.

To get an idea of what old-time music sounded like in the 1920s it’s very easy to find some of those early recordings online. Some of the first people to ever record old-time music were Fiddlin’ John Carson, Henry Whitter, and Samantha Bumgarner. As another example take the lyrics from the popular song “Devil Went Down to Georgia” which describes the traditional old-time music Johnny was playing in the song. The line "fire on the mountain, run boys run" refers to the traditional fiddle tune "Fire on the Mountain". "Devil's in the House of the Rising Sun" probably refers to the folk song "House of the Rising Sun" (the best-known version of which was recorded by The Animals). "Chicken in the bread pan pickin' out dough" is what they call a "floating verse" which is basically just a filler line that's found in various folk songs. And "Granny Will Your Dog Bite?" is the name of another fiddle tune. So he's basically just playing some old-time standards that any random hillbilly fiddler from Georgia would know.

Recording companies in the 1920s recognized the demand for this type of music and marketed it with a variety of names that invoked a sense of nostalgia or an old-fashioned quality. Record companies like Victor, Columbia, and Okeh embarked on recording sessions throughout the South to attract rural folk musicians. An advertisement for a major recording session in Johnson City, Tennessee asked "Can you sing or play old-time music?". Country historian Bill Malone wrote:

As the number of rural bands and musicians proliferated in the mid-twenties, observers, friendly and otherwise, were hard-pressed to find a name which correctly described the kind of music they performed. Record company release sheets and catalogues camouflaged the songs under a variety of headings, most of them evoking a romanticized conception of southern rural music. Interestingly, no one ever described the music as "country," although a term like "hill country tunes" might sometimes be used. The presumed "southernness" of the music was often evoked in accompanying illustrations showing happy "darkeys," doleful or feuding mountaineers, or cotton fields, and early Brunswick catalogues described their recordings as "songs from Dixie." The folkness of the music was dimly recognized in such labels as "old familiar tunes" (used by Columbia) or "old-time music" (favored by the Okeh label and by many fans and musicians). The word, though, which was destined to crowd most of the romantic terms aside, and which has persisted to our own day, was "hillbilly."

I won’t get too much into this here, but “hillbilly music” was a sort of catch-all industry term to refer to any white rural music in opposition to “race records” which was used to describe any black music, despite the fact that old-time music was and is a shared black and white tradition. Another user noted Henry Ford promoted the quadrille and reel as older dance forms. Ford promoted square dancing as a pure American (read: white) opposed to other forms like jazz. Here again it’s important to note that square dancing was and is practiced by black communities as well. Regardless, the supposed rural or “folksiness” of old-time music has been at the center of its commercial promotion. The early radio shows that featured old-time music also evoked this sort of down-home images. There were many radio shows called “barn dances” in the 1920s that featured country music. The most well-known of these is the Grand Ole Opry which started out as a barn dance program, but it’s moved on from strictly traditional or folk music.

Many old-timers who grew up in the tradition say that square dancing and old-time music nearly died out with the rise of other popular genres like rock & roll. Thankfully there was a resurgence thanks in part to the larger folk revival in the 60s. Even though it’s been called old-fashioned old-time music is still going strong in many parts of the country, among fans and musicians both young and old. You can see old-time music played at huge community events like fiddlers conventions in Galax, Virginia or Mount Airy, North Carolina.

I don’t want to say much more so I’ll just list some references and sources.

Early old-time and the recording industry:

Country Music, USA – Bill Malone and Jocelyn Neil

Tennessee Strings – Charles Wolfe

Black influence in old-time music and dance:

Sinful Tunes & Spirituals - Dena J. Epstein

African Banjo Echoes in Appalachia - CeceliaConway

Hoedowns, Reels & Frolics - Phil Jamison

Regional specific studies of old-time music:

Play of a Fiddle – Gerry Milnes

Blue Ridge Music Trails of North Carolina – Fred Fussell

There’s tons of great documentaries on folkstreams.net and there’s even /r/oldtimemusic .