Do they have any practical use, outside of destroying infantry?
That is just a myth.
But they do affect the weather on the other side of the globe.
Shake the foundation of the Empire State Building.
Grow tomatoes as big as watermellons.
What ever else you would like.
Tesla Coils are good for:
1.) Practical jokes – point it at doorknob when someone’s grabbing the other side of the knob.
2.) Science demnstrations – create “lightning”, use it to make fluorescent tubes glow
3.) A real use – you can “tickle” a gas discharge tube with a Tesla Coil. a Good wa to get a recalcitrant laser tube going.
Tesla also reportedly lit up his entire labritory wirelessly with a big coil in the center, and neon bulbs stretching out from it.
Even this isn’t practical, because they suck against tanks. And who’s to believe that you’re going to be attacked with just infantry?
glaiven
From what I’ve read, Tesla’s aim with the invention of his renowned air core transformer coils was to create a means by which electrical power could be transmitted without the need for physical power lines. In that sense, you could say he was a bit of a mad scientist … supposedly he envisioned a world where future cities could use giant inductor towers instead of electrical lines for both communication and the transmission of power. This does not mean he wasn’t a brilliant man, of course, since his research taught electrical engineers of the day a lot about the limits of electrical air conductance and why it wouldn’t work in that capacity.
On the other hand, Tesla’s induction coil was essentially an ancestor to the more refined ignition coils that we use today for the engines which power our cars. I believe that similar transformers also used to be implemented in CRT displays due to the high voltage needed to run them. I suppose they still are, even though CRTs are quickly becoming relics of the 20th Century. I guess the best thing Tesla coils are good for is to serve the interests of electrical enthusiasts who like to build them and bring them into coil conventions to see which one can make the biggest arc of electricity. 
Hm, my reading had it that his intention was to send power through the earth (changing “ground” to charge) to light cities and do this for free. His tower, though, might have been created for communication purposes (radio, telephone, etc.)
As to the use of the coils–I don’t know electronics (or, specifically, electricity) all that well, but Tesla held his coil as being the ultimate electric doohickey for all purposes practicle or simple research. Unfortunately, he could get a standard coil to do various magical things while as others had to fiddle with it to get it to do any single thing But, he never taught anyone else how to do any of the stuff he did and these days, those things might not be terribly impressive.
Sure, We use one all the time in our lab for a practical, commercial, yet proprietary reason. I could tell y’all, but then I’d have to zap ya.
http://www.tlchm.bris.ac.uk/safety/vacuum.htm says small Tesla coils are used “to test for pinhole leaks in glass vacuum systems but also for testing insulating coatings, reming deposits from insulators and exciting discharge lamps.”
http://www.violetwand.org/tesla.htm says that appliances called violet wands are modified tesla coils. Violet wands were originally an electrical snake oil intended to cure cancer and stuff. But now… well see this site: http://www.violetwand.org/home.htm
Does this count as a “practical use”?
The flyback transformer in most television sets and monitors is not a Tesla coil, but it does share some design similarities. IIRC, a Tesla coil used an air-core transformer whose primary was driven by high-frequency electric pulses coming from a spark gap and capacitor circuit, to generate a high-voltage, high-frequency output. A flyback transformer is driven by a solid-state switching circuit instead, includes a rectifier to turn the output into high-voltage DC, and has multiple secondary windings to generate additional voltages needed by the television. It’s actually not hard to build a Tesla coil out of a flyback transformer and a few additional parts.
Moreover IIRC the flyback transformer uses the television’s horizontal sweep frequency with its very rapid retrace or “flyback” to generate high frequency components that help push power through the tiny transformer - and they also used this horizontal frequency as the transformation frequency to generate lower voltage used in the audio amp - three different purposes funneled through this one part.
Which, by the way, was a solid core transformer (I guess ferrite or maybe powdered iron).
And as noted the Tesla core is an air core device.
BTW he pronounced his name “Tezla” though many people today say it with a hard “s”.
The last of a generation of magical inventors, he was.