It's a percussive drilling system, or a cable-tool drilling system. Here's a chart. Basically, it's a metal rod that is suspended on cable, and repetitively raised and dropped, driven by an engine. In that era, it was likely to be a steam engine of some sort, although in the early 19th century one could use beast or human power, too.

Periodically, a basket is lowered to pick up the cuttings, and men have to go down the borehole to pick up larger pieces. Mud is often put into the borehole to soften up the soil.

You can read more about it on this basic website or this one. In that era, one could only drill vertical holes using cable drilling. Since the early 20th century it has been replaced by the rotary drilling system.

While cable drilling is no longer used in oilfield, it is still used in construction and landscaping, and for drilling water wells. The next time you happen to walk near a construction site (or worse, work in an office near one), the BAM---BAM---BAM---BAM noise that you hear is the modern version of the cable or percussive drilling system.