In the 1960s and '70s, when I was growing up, the word burlesque was synonymous with strip club. Perhaps a higher-class sort of strip club, but nonetheless a sordid place of crass entertainment.

In 1900, when
Joe Weber and Lew Fields were operating their Broadway Music Hall,
burlesque still held its traditional meaning. This is how
Wikipedia defines it:
Burlesque is a literary, dramatic or musical work intended to
cause laughter by caricaturing the manner or spirit of serious works, or
by ludicrous treatment of their subjects.
It differs from satire in that satire uses humor to illuminate some truth, while burlesque pokes fun for the sake of fun. It's caricature without a chip on its shoulder. The book
Catch-22 is a satire of the military in World War II. The TV show
Hogan's Heroes was a burlesque. In the 1960s, television was the medium of choice for burlesque (though the word was never used to describe it).
Get Smart, F Troop, and
Car 54 were burlesques of the spy, western and police genres. More recent examples are the films of Mel Brooks, Leslie Nielsen, and the team Zucker, Abrahams and Zucker. The situation comedies of today's TV all try too hard to have some point--even shows like
The Simpsons and
South Park feel a need to preach.
Burlesque first became popular in New York when Lydia Thompson arrived with her troupe, British Blondes, in 1868.
Thompson's Wikipedia entry includes this quoted description:
"The eccentricities of pantomime and burlesque – with their curious
combination of comedy, parody, satire, improvisation, song and dance,
variety acts, cross-dressing, extravagant stage effects, risqué jokes
and saucy costumes – while familiar enough to British audiences, took
New York by storm."

The burlesque of 1900 was mainly about making fun of some current trend or topic. In New York, Weber and Fields were the masters and their eponymous music hall the most popular venue. A typical show included songs, chorus girls, and several loosely connected scenes and skits. Weber and Fields specialized in burlesques of contemporary Broadway theatre.
The images I've used here are from the program for the week of October 15th, 1900. This included their immensely popular piece
Fiddle-Dee-Dee. This piece is set in Paris at the time of the
Exposition Universelle of 1900. The hero, played by
DeWolf Hopper, is described as "an athletic young American, with nothing but money and nothing to do but spend it." Apparently, he meets up with a "Hebrew prestidigitateur," or conjurer. Weber and Fields play supporting roles, and the female leads are played by
Lillian Russell and
Fay Templeton. The latter had a big hit with the song
“Ma Blushin’ Rosie, Ma Posie Sweet.” Norman Hapgood provides some further description of
Fiddle-Dee-Dee, and Weber and Fields, in his book
The Stage in America.

Also included in the show was an "incidental" dance by
Bessie Clayton, "La Danse d'Afrique." Bessie Clayton, a regular at Weber and Fields, was apparently trained in ballet, and
one article describes her as an Amercian
Genée. According to someone knowledgeable in the field, she was also
the mother of toetap dancing. Altogether, there are about fifty cast members listed in the program. From what I can tell, many of the same performers were with the company throughout the life of the Music Hall (1896-1904).
The second "exhibit" was a spoof on the current Broadway production of
Augustus Thomas's Arizona. The same company appeared, this time with Hopper playing Henry Cannedbeef, and Joe Weber appearing as Lena Killer. At other times that season, the second exhibit was
Quo Vass Iss?, a takeoff on the Broadway show
Quo Vadis, which was based on a popular novel set in ancient Rome. Hapgood provides some of the lines:
I have just returned on the limitus vestibulus from Asbury Park.
You must have a thirstus fit to float a galley. Thou art an easy Markus.
Let us to the boozeorium.
It's unfortunate there are no recordings of these shows, but there are some of Weber and Fields' later works. These are
vaudeville routines recorded in 1912 and 1915. I really enjoyed listening to these. Their playful manipulation of language brings to mind the Marx Brothers (the contract scene seems very similar to the one in
Night at the Opera). But of course, Weber and Fields did it first.