As for running vs left on, an appliance or piece of equipment can be left on but not running – if it's not been powered off, but it's not doing anything, either.
For example, consider my DVR: I can leave it on all day, but it would only be running if it was recording, or I was watching a movie.
The terminology can vary slightly depending on how the device is designed. If you can power a device on or off without it doing anything else, then it can be left on even when it's not running; examples would include a cell phone or a computer. However, some devices can't powered on unless they are actually doing something, like a toaster. You can't simply leave a toaster on.
Fans can be turned on, but, once they are on, they are running. You can leave a fan running, and you can leave a fan turned off. If you leave a fan turned on, though, the implication is that the fan is running – but that wouldn't hold true for a monitor.
And some devices don't really run at all. For example, I can't imagine myself saying, "the radio is running." It's hard to state a general rule that will work in all circumstances, but I would say that if the device has a motor, then it runs, and if it doesn't have a motor, it doesn't run. That's why my DVD player and my refrigerator both run, but my television and my oven are simply on.