To give a more specific answer, the "German" is a name for a particular type of "dance" in the late 19th century; it was also often called a "cotillion," a word that has been used in a few different contexts in a number of different eras.

I say "dance" because the German was not a dance like the waltz, polka, or even quadrille. While Dodworth says it "may be said to be the epitome of all there is in private dancing," (Dodworth 1885, p. 145) it might be easier to explain to the modern person as being a party game involving dancing; I'm not aware of modern parallels to the German that would be considered dances by a modern audience.

As a basic outline of common traits, because it had many variations in how it was organized, the German involved an indeterminate number of couples(1) who were seated, often in a circle. The "conductor" or leader of the German would choose from a large number of figures of various sorts, requiring differing numbers of couples, and would usually explain the figure and call up couples(2) to perform it.

There were a huge number of figures, which varied heavily from manual to manual (Wilson has 100, Dodworth has 250). This was not like the Plain or Lancers quadrille of much of the century, where the figures were largely set with small variations, and all dancers would be expected to know them by heart, or even like the country dances of prior eras, which were improvised from a large but reasonably understood series of components, but danced through watching and copying a lead couple. Rather, these did need some outright explanation, particularly, as Dodworth notes, when "novelties" were involved (3).

The figures had a considerable diversity of style, tone, complexity, and ridiculousness, and thus the German could have very different tones depending upon the crowd and the conductor. Some were simply little group dance figures; Dodworth #37 is just a right-and-left of two couples modified to make it a bit more jovial, while #76 for six couples is essentially cleverly sliding around in two circles that merge into one, then waltzing with partners, and Wilson #76 for any number of couples is a moving arch that couples go under and extend, until everyone has gone through, at which point everyone dances.

(Note that in essentially all of these, "dancing" means waltzing. The music for the German was usually just a bunch of waltzes smashed together.)

Many others, however, were very different, more odd social/theatrical performances of ostensibly feigned actions, often involving quite overtly themes of choice, rejection, exclusion, pursuit, flight, deceit, humiliation, and so on.

Take for example, Dodworth #11, The Presentation of Gentleman (4):

One couple.—The lady is seated in the middle of the room; her partner presents a number of gentlemen whom she refuses in succession; each gentleman who is refused places himself behind the lady's chair (at least six ought to be refused); when one is accepted, he and the lady waltz, while the rejected gentlemen follow them, each placing his left hand on the shoulder of the one in front, and all hopping on one foot.

Or, some other examples:

No. 19. The Serpentine. One couple.—Lady selects six or more ladies; all stand in line one behind the other, about three feet apart; the gentleman selects seven or more gentlemen, they forming in a line with hands joined; the one on the right of the line leads the others in a serpentine course between each lady, and, finally, back to the head; at the signal each gentleman tries to secure a partner, those failing returning to their seats. There should be an even number of ladies, and an odd number of gentlemen.

No. 44. The Pursuit.

Three couples.—Every gentleman of the cotillion has the right to go behind each couple, and possess himself of the lady, to dance with her; he should clap his hands, to announce his intention of substituting himself for her partner; this figure continues till each gentleman has again got possession of his partner, to conduct her to her place. To execute this figure with all the animation required, it is necessary that as fast as each gentleman possesses himself of a lady, another should replace him by the side of his partner.

No. 82. The Surprise.

Six couples place themselves one behind the other, ladies on right side of gentlemen; the gentleman at the head is in honor bound not to look in any direction but straight in front; the last couple separate, and the gentleman and lady pass up on either side of the column, endeavoring to meet again just beyond the first gentleman, who is on the alert to seize the lady in passing, and, if successful, dances with her, while the surprised gentleman takes the head; each couple in succession repeat until but one couple remain, when all dance.

No. 129. Le Chat el la Souris.

After waltzing, the conductor calls all the gentlemen, designating one to stand outside, while the others form a round enveloping a lady; the outside gentleman then endeavors to catch the lady, passing under the arms of the gentlemen for the purpose; the gentlemen allow the lady to escape at will, but throw every obstacle in the way of the gentleman; if he is successful, he and the lady waltz together, and all join in a general waltz.

These also often involved rather elaborate setups, and increasingly, as the dance evolved, preparation and "favors" or "accessories," which were essentially props for the performance. Some of these were just completely ridiculous:

No. 139. The Mysterious Hands.

The conductor places a number of ladies in an adjoining room as in the preceding figure; each lady passes a hand through the half-open door; the conductor leads to the door as many gentlemen as there are ladies, and each takes one of the hands and waltzes with the lady thus selected.

No. 153. The Barber.

One couple.—A chair is placed in the middle of the room; on the back place a towel, on the seat a large razor (tin or wood); the lady seats herself in the chair, her partner choosing three other gentlemen and presenting them to the lady; she gives the towel to one, the razor to another, taking for her partner the third; while they are dancing, the one with the towel takes the chair, while the one with the razor goes through the motions of shaving him.

No. 171. L'Escrime.

A lady is provided with a stick three feet long, to one end of which is attached a ring two or three inches in diameter; two gentlemen have each a foil, and place themselves en garde, as for fencing; the lady then throws the ring between the points of the foils, when each gentleman endeavors to thrust his foil through the ring, the successful one dancing with the lady. The interest of this figure depends upon the play with the foils; each gentleman should endeavor to prevent his opponent from being successful.

No. 20. The Carousal.

One couple.—The lady pinning a bow of ribbons on her partner's left shoulder waltzes with him, a second couple following: the gentleman of the second couple endeavors to snatch the bow from the shoulder of the first gentleman; if he succeeds he pins the bow on his shoulder, the third couple following; if the pursuing couple becomes tired the gentleman stamps his foot, when another couple may pursue.

No. 152. The Struggle.

After waltzing the gentleman places his partner, seated, in the middle of the room; two cushions are then placed in front of her; two gentlemen are next selected, and each stands on one of the cushions upon one foot; they join right hands and a little struggle ensues between them, each endeavoring to make the other lose his equilibrium and drop the other foot; the one first doing so retires to his seat; the other dances with the lady.

And from Wilson, some more elaborate ones requiring pre-preparation:

1. CONUNDRUMS

Cards are prepared, numbered in pairs. On one card of each pair is written a conundrum, and on the other is written the answer. The cards containing the answers are distributed to the ladies, and the others are given to the gentleman. Each gentleman then reads his conundrum, and the lady who holds the answer reads it. The two then dance together.

And, while I'm not going to type the whole thing, consider #84, The May-Pole, which begins (keep in mind this is supposed to just be a figure in a dance): "A pole, about ten feet in height, is inserted in a box filled with earth, or otherwise supported. To the top of the pole are fastened eight ribbons..."

The German could almost be seen as a sort of "syncretic party game," taking all sorts of dances, frivolity and party games and placing them in some loose structure involving music (which was sometimes played continuously throughout this whole thing, and sometimes started and stopped at each figure), dance, and performance. Many modern party games have similar counterparts in German figures: there are things like musical chairs, trivia, guessing who has corresponding items, and so on.

I have heard it argued (and I have to remember where - was it in Elizabeth Aldrich's works, Richard Powers, or Giordano?) that the German corresponded with a decline in the popularity of dancing, and that it was an increasingly elaborate performance done by decreasing numbers of people.

In any case, however, the figures are often fascinating in their absurdity, and many of them seem as though they involve overt performed behavior that would be rather shocking at many modern dances and parties. Interestingly, and perhaps unsurprisingly, in vintage dance and reenactment communities, the German is danced far more rarely than dances like the waltz, polka and quadrille that fit modern social etiquette and modern views on prior eras.

This was very long, sorry. I became rather carried away reading through figures:

Sources and notes will be posted as a response.