Somehow, as I was cleaning the drain, my finger got stuck in the little trap part right inside the drain. I evidently went in at just the right angle such that I couldn't get it out. We tried a few things but it came down to breaking the sink to pull out the trap and then pushing on the top of my finger to get it out. At least we were able to do that. It would've been embarrassing to go into the emergency room to get a sink trap removed from my finger.
In the future, I'll try to be a more tool-using monkey.