In the 1890s, your photographer would certainly be using glass plate negatives (negatives on film had only just been invented, and were really only being used in amateur snapshooting). The camera would use plate holders for the negatives--before shooting, you load the glass negative into the plate holder in a darkroom (photographers (or their assistants) would do this in advance and take a bunch of loaded plate holders out for the shoot). When shooting, you set up the shot without the negative in the camera. The back of that kind of camera (a view camera--http://en.wikipedia.org/wiki/View_camera) is a big piece of glass the size of the negative you're shooting with, through which you see the scene from the lens, but upside-down (
http://en.wikipedia.org/wiki/View_camera#mediaviewer/File:Sinar_F_-_visée.jpg). To compose and focus you open the lens and use the ground glass. For better clarity you might put a hood over your head and the camera, as you've probably seen. Once you're ready, you close the lens and put in the plate holder (it goes between the bellows and the ground glass). The plate holder has a thing called a "draw slide," which you pull out so the plate will be exposed to the lens when it opens. You then click the shutter, which opens the lens (if using flash, you set it off at the same time--I think you'd certainly need an assistant to do one or the other).
Development of glass plate negatives is not that different from film (well, dry-plate glass negatives, which is what you're using). You do need a darkroom. The plate then goes through the baths of developer, stop bath, fixer, and wash. At least an hour start to finish, although once it's in the fixer you turn on the light to look at it. It then needs to be completely dry before you print with it--my guess is at least an hour or two for that.
Glass plate negatives are contact printed--no enlarger. Glass negatives are large--there's variety but for your kind of thing maybe around 3x5 to 5x7 (inches). To contact print, you need a contact printing frame (
http://www.lotusviewcamera.at/accessories/contactprintingframes_e.html). You load the negative into the frame, and then the paper behind it, and close it very tightly. You then expose it to light. The kind of paper you'd be using would be printing-out paper (silver gelatin or collodion), which is much slower than the kind of paper you'd use with an enlarger. You expose it using UV light--ie, sticking it outside facing the sun or in a window. On a sunny day it would probably take 20-30 minutes to expose. You then take out the print--which has a visible image, unlike the paper you use with enlargers--to develop. For printing out paper, the first step is washing, followed usually by toning (not neccessary, but usual), then fixing, then wash, then dry. Again you can turn the light on by the fixing stage, and you can see the picture as soon as you take it out of the frame. About an hour from the first bath to the end of washing, then a couple to fully dry.
Whew! That might be wayyy more than you need, but that's the overall process. If you do want more details, let me know--I've never shot with glass negatives but I've used an old view camera and developed photos with printing-out paper, including one from a glass negative, so I could give a more detailed run through of the steps. I am actually a photohistorian, with a master's degree and everything, so if you have any other related questions, I might know or would know where to look at least.
Also, you might have already seen these, but for the kind of photos that might be taken indoors with flash at the time, have a look at the work of Jacob Riis, from the late 1880s and 1890s:
http://www.moma.org/collection/artist.php?artist_id=4928