A mix of plant extracts and human urine, and animal fluids. Generally, basic (alkaline) solution was needed to soften the hide and make it easy to remove the hair. I'm not an expert on this but, the National Park Service has a good summary on this, as part of the page on Jamestown National Historical Site..

Here's the relevant passages:

The first vat contained a lime solution (obtained from limestone or shells), into which the hides were placed and left to soak, hair side down. Chemical action caused some of the hair to fall off. The skins were then removed and the bleached fat and hair-lime residue was scraped off, the latter later sold for plastering. The hides, scraped free of the bulk of the hair, were returned to the liming vats, which by now contained large quantities of decaying organic matter full of bacteria which aided the process of hair and connective tissue removal. In a procedure which took about two months, the skins were alternately soaked for a few days and removed from the vats, folded and left out for a few days more. By the end of this process, the remaining hair could be easily removed without harming the hide.

The skins then entered a similar soaking and resting period, involving vats of lime in weak solution. This process took over six months.

In the next phase of leather production, the hides were combined with oak, beech or willow bark, which would provide the tannic acid necessary in preserving and coloring the hide. Other acidic additives to this dressing included sour milk, cider pressings and ferment of rye, recipes varying according to the materials at hand, the weather and the ultimate finish desired. Bark was spread in the bottom of the tanning pit, then hides and bark alternately stacked until the pit is full. A foot of tanbark covered the pit, and the whole well trampled down and kept moistened for three months.

Unpacking the pit was a risky business, because the hides were very soft and vulnerable to tearing.

To stop the action of the acetic solution, alkaline dressings were then applied. Ingredients included soft soap, boiled meal and dog, pig or fowl dung.

The final step in leather preparation was performed by the currier, a specialist who worked the leather with oils and greases, using a variety of tools to prepare the leather for its intended use.

There was also a practice of using the fluids from the brainpain (cerebrospinal fluid?) to tan the hides, but that is a practice that was usually practiced more in the backcountry, and I know less than nothing about it except for a few old jokes.

However, I just realized that you were asking about the late 1800s, not the late 1700s. By the late 19th century, the essential process would be similar, but they would probably use slightly different sources of the ammonia and the tannins. In particular, by the late 19th century, the use of synthetic chrome salts as "dyes" allowed for much more color-fast blacks in the leather, as well as a wider range of colors, including blues, reds, and purple that had previously been difficult to obtain.

Here is a link to a PDF by a local historical society describing the 19th century leather industry in rural Connecticut. The process is essentailly similar, with the use of limewater and ammonia (from urine, probably) as alkaine agents, and tannins extracted from hemlock or oak bark as acids agents.