The Jacob's Ladder is a high voltage climbing arc
Quick Physics: The transformer at the bottom creates a potential difference between the wires. The electrons repel each other, so they jump from one wire to try and get as far apart as possible. The spark heats up the surrounding air and hot air rises, so the spark rises with it. When the spark gets to the top of the wires, it dies and a new one starts at the bottom.
The Jacob's Ladder is a relatively simple device. The big box on the bottom is called a transformer. A transformer is something that changes the voltage going to a device. You probably have several transformers in you home; for example, the charger on your cell phone is a transformer. Your cell phone converts the 120 Volts that come out of the wall into 9 or 12 Volts. The Jacob's Ladder converts the same 120 Volts to more than 500 Volts!
When the Jacob's Ladder is turned on, electrons are fed into one of the wires. These electrons want to get away from each other, so they jump across to the other wire, which is connected to the ground. When they jump, we see a bright spark in the air. The spark then climbs up the ladder as it heats the air around it. Remember that hot air rises, and in this case takes the spark with it. This spark is very hot, so hot that it can be classified as a plasma (see Plasma Tube). Eventually the spark dissipates and releases all of those electrons into the air.