monotype -- several patents from 1885-1890 covered this system and its basic operation for ordinary text. the wikipedia article covers this pretty well. what it doesn't mention is the "4-line system" that was required for math. this is described in daniel rhatigan's m.a. thesis . this is the system used to compose the first editions of the first two volumes of knuth's "the art of computer programming", and it was the switch away from monotype that is the reason that tex exists today.

the ibm selectric and "clones" used a "golf ball" mechanism. these type units were easy to switch, although only one "font" (containing the same number of symbols as keys on the keyboard, times however many shift keys were provided) could be in use at one time.

the "varityper" , had a revolving drum onto which two semicylindrical fonts could be mounted, allowing both latin and greek (for example) to be available at the same time. switching one font for another was reasonably fast, and the use of proportional spacing resulted in an appearance that was more refined than possible with a monospace typewriter, but not nearly the quality of real composition.

traditional one-symbol-per-key typewriters. an "ordinary" typewriter has keys linked to type bars containing usually two symbols, with a "shift" key to change from lower to uppercase. any symbol not on one of these type bars had to be added by another method, either by hand or by the use of a widget called a "typit" , a plastic device with a raised symbol at one end that could be stuck into a guide mounted on the typewriter where a key would impact the ribbon, and any key could be struck to impact the typit and transfer the image of that symbol to the ribbon and paper. typits were used on machines with either proportional or fixed spacing.

"typewriter" composition. actually, this is an oversimplification. the "direct type" methods were of two distinct kinds. both were best used on machines with a carbon rather than an inked ribbon.

photocomposition was the next advance. i know of one direct-input photocomposer used for math -- the photon 200. a keyboard with several automobile-like shift levers controlled the input. images of symbols were transparent shapes on a glass disk otherwise coated with black, like a photographic negative. for "pi" symbols not on the disk, there was a "whirligig" with eight arms, onto each of which could be mounted a photographic negative of one symbol; jobs had to be carefully edited beforehand to make sure that no more than the eight available pi symbols would be needed, since changing the pi set in the middle of a job could run the risk of exposing the output medium, which would mean starting over. there was sufficient computer power in this machine to calculate line length and apply justification (in text), but for the most part, such decisions for math were manual, and required a highly skilled operator.

for more automated photocomposition, input was keyed into some recording medium -- punched cards, paper tape, ocr forms, floppy disk, ... -- and input to the typesetting machine for output to photographic paper. since photographic paper was messy and expensive, there were sometimes preliminary proof runs in which the input code was printed out on greenbar; not ideal, and accurate really only for text, but often less expensive than other alternatives. (and patching paper tape was definitely an "interesting" undertaking.)

output devices varied in how images were transferred to the photographic medium.