Edit: I figured it out! Right click the taskbar, "Add widget" and select "task manager."
The applications I have open don't appear in the taskbar. My taskbar is just a big, empty bar at the bottom of the screen no matter how mamy applications are running. When I minimize something, it just vanishes until I use alt+tab to reopen it.
I had right-clicked the taskbar and was messing around with cosmetic changed and apparently did something irreversible. The problem started, I think, when I removed a panel, or maybe I removed a widget from a panel. I don't even know what to Google to figure out how to resolve this (and Lord knows I've tried). It was so easy to change that setting in the first place and I cannot for the life of me figure out how to undo it.