Okay, nobody get excited. It was basically the least complicated possible fix -- but I know, like, 15 times more about how sinks work now than I did ten minutes ago. Caitlin is washing her laundry in the sink -- we've been putting off laundry, and now the fact that people in other apartments are using the washing machines is an unacceptable obstacle. When she was getting started, she pulled on the drain rod to close the pop-up drain stopper[1. Look at all these words I know now!] and it came out. Completely.
Now, the drain was shut, so she was able to start doing the laundry. But we both felt that we had not arrived at a permanent solution.
So, I employed my vast experience of internetting to google the solution. I found someone who calls herself Pretty Handy Girl, who fixed her mother's sink. Her mother's sink was much more broken than ours.
Via that explanation, I learned enough about how sinks work to figure out that the vertical drain rod had slipped out of the plastic bit that seems to be a part of our sink but not sinks in general that connected it to the horizontal rod. From there, it was a matter of re-attaching it and tightening the screw that held it in place. It was really, ridiculously easy, and I had a surprisingly hard time doing it. Being under the sink hurt my back and arm. I may have accidentally elbowed Caitlin in the eye.
But our sink works. And when it breaks again in a couple days, we'll know how to fix it again. And when it breaks again after that, we'll know how to fix it and that time we'll probably bother to use a wrench and tighten the screw properly.