This shit is MYSTERIOUS.
So basically, older bikes have "fricton shifters." The lever moves seamlessly, tightening the cable and moving the derailer across the gears. Tighten it, and the cable pulls the derailer towards the bike, moving the chain onto another gear. Pretty simple. Lately I've had issues, and a weird thing started happening with my lever. Now instead of moving seamlessly, it "clicked" between positions. Except it isn't actually moving cleanly between gears, it is putting my chain awkwardly in the middle.
I looked it up, and apparently I have a shifter that is an intermediate step between old-fashioned shifters and the modern ones that have little numbers and stuff. So I spent a while poking around looking at schematics, trying to figure out how to set this thing up. There is a little red arrow on the side that seems to indicate that SOMETHING can rotate to set it up properly-- there is a corresponding arrow on another part that says "SET" and another that says "FRICTION" and "SIS." I couldn't figure out how to move any part of it, and couldn't figure out what model I have.
So I finally just tried adjusting the derailer (the little mechanism that moves the chain around). I tightened a screw (the high-limit screw on the derailer, for you guys following along at home), and now everything works, and my shifters don't "click" anymore. So the problem is solved, but the ways of the bike are still a mystery to me. There is definitely still a weird clicking sound coming from the lever intermittently, but I don't really know how to address that.