I have an old Panasonic 8mm tape camcorder. If it is plugged in, it can record, playback video, and everything. However, if it is not plugged in, it simply won't power on. What is the problem here?

Typically, if the camcorder only stays on when plugged in, the camcorder battery is no longer good. Batteries wear out over time and eventually need to be replaced when they stop holding a charge.