You'd find, really, two kinds.

The most common by far would be hide glue, essentially animal collagen. That came ( and still comes) in various different kinds. Someone assembling furniture would have a big pot of regular hide glue kept hot . But a gilder or bookbinder would use lighter rabbitskin glue, which tended to bond quicker. And there was Isinglass, or fishskin glue, that didn't gel quickly, like regular hide glue would when cooled, and so was supposed to be the strongest.

The other kind would be casein glue. Previously, this was called cheese glue, because that was the basic ingredient. Cheese ( with as little fat as possible) would be treated with lye and quicklime. It had been used for joints that needed water resistance and more strength: if a 16th c. painter did a portrait on a wood panel, the pieces of that panel would often be assembled with cheese glue. But sometime in the earlier-to-mid 19th c. making the glue became an industrial process ( I think Swiss) and it became more common. One reason for the small surge in the popularity of hammered dulcimers in the US circa 1850 seems to be the stuff was more reliable, under the heavy load of the strings, than hide glue. It also found use in the piano industry and in making plywood.

There would also be some clever concoctions involving raw rubber, pine rosin, shellac,, various resins, even tar...but these would be for something like sticking together a broken jug or gluing felt to a desktop. Hide glue or casein would be used for furniture making.

While casein has mostly been displaced by urea-formaldehyde glues, hide glues are still quite common. It can make a glue joint that's almost invisible, and its lack of water resistance means pieces can be disassembled with a little heat, water, or steam: without that feature, all those priceless violins would never be able to be taken apart and restored.