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Put in other words, the plasma is a greedy conductor of electricity. Suppose it has just enough free electrons to get a current started. The current causes ions and electrons to move rapidly and to collide violently, and those collisions strip additional electrons off atoms of the gas. Additional electrons increase the current, causing more collisions and producing still more electrons, which create more current, more and still more... In this way, if a fluorescent lamp were directly connected to the power lines, unprotected, its current would rapidly grow until something gave way. The tube might heat up and explode, the wiring might melt... or more likely, the fuse or circuit breaker which protect the fixture would stop the current.
The Ballast
A resistor connected in front of the tube, in place of the ballast coil in the drawing, would prevent this from happening. Imagine our power comes from a 110 volt line, and the resistance in front is 220 ohms: then even if the effective resistance of the plasma falls to zero (and it can't fall any further!), the current drawn is only (110volt/220 ohm) = 0.5 ampere. If the plasma adds its own non-zero resistance, that makes the denominator larger and the current even smaller.
Why then a coil and not a resistor? Because the tube is fed by an alternating voltage, which rises and falls 120 times a second (in the USA; 100 times in Europe). Its electrical current sloshes back and forth, 60 times a second in one direction, 60 times in the opposite one. In between, 120 times each second, the voltage drops to zero and the tube is extinguished, since plasmas react very quickly. Somehow, it must be relit!
A ballast coil can do that. In an alternating current, it acts a bit like a resistance. As the current rises, it absorbs energy from it to build up its magnetic field, slowing down its growth. Then, when the voltage drops to zero, the stored magnetic energy produces a voltage surge which relights the tube. You will not usually see the fast flickering of the light, except maybe if you illuminate a rotating fan, when (at the right speed) its motion seems to stop. (Note: compact fluorescent lamps now exist in which the ballast coil is replaced by a more complex electronic circuit. The flow of electric current is then limited by transistors.)
And what about this "fluorescent" thing? The mercury atoms in the plasma generate light very efficiently, but much of it is ultra-violet (UV), invisible to the eye and harmful to it (or rather, it would be, were it not absorbed by the glass). The solution is to coat the inside of the tube with a glow-in-the-dark (fluorescent) paint, which absorbs the UV and re-emits its energy as visible light.
All other plasma lamps--sodium and mercury streetlights, neon lights etc.--require ballast coils, too. Recently, small fluorescent lamps have appeared on the market, which screw into the socket of a regular lighbulb. They have transistor circuits replacing the coil, and although they cost more than filament lamps, they are (like other fluorescent lamps) much more eficient.
(And if you think Ohm's law is badly violated by fluorescent lamp plasmas--just wait till you read about the ring current, the electric current carried around Earth by trapped ions and electrons of the radiation belt. That current needs no voltage at all, it circulates just because of the trapping of the plasma!)
A few words about safety
If a fluorescent lamp were not protected by a ballast, it could in principle draw a huge current. Occasionally (not too frequently), a ballast coil fails badly, the circuit breaker fails to do its job and a fire is caused. The usual sign of a failing coil is a loud hum from the fixture. The reason: to prevent parasitic currents, the coil is wrapped not around a solid iron core, but around a stack of iron plates, insulated from each other by a tar-like substance. On some old fixtures, those plates work loose and start vibrating at the frequency of the alternating current, which to our ears sounds like a deep hum. Violent vibrations may scrape the coils wrapped around them and allow the plasma to carry a greater current.
A low-intensity hum is probably no cause for alarm, although it can be annoying. But if the hum gets really loud, it my be safer to replace the coil or the fixture. Electric transformers are also constructed around stacked iron plates and are subject to the same problem.
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