Colloquially, any technology designed to use electricity and a heating coil to vaporize a solution of propylene glycol and/or similar substances in order to prodce a cloud of breathable water vapor. The form factor can be anything from an e-cigarette to a full on box mod, but the gist is to replace the smoking of tobacco or other substances with healthier vapor. The glycol is typically mixed with flavoring agent, to which nicotine or other substances are added, simulating smoking.
In essence there are three parts to a vaping setup - you have the "tank" or clearomizer - basically a holding receptacle for the juice and a strawlike tip for inhaling the smoke. These may vary wildly in appearance but they usually have at least a window to see your juice level. The coil is the part that has some kind of wicking device to draw in the juice by capillary or other action, and then heat suddenly with an electric charge to vaporize the juice against it. This is fed by any of a number of control units - the simplest of which is a battery and a switch, but can get as complex as a box the size of a pack of cards with a screen readout telling you wattage and battery power level, along with temperature and voltage control.
The components are threaded with standard threads, meaning that in terms of physical compatibility most of the units will bolt together and seal just fine, but it's best to be kitted out at first by a bona fide shop. Putting a sub-ohm tank on an e-cig doesn't make much sense, and a clearomizer on a box mod would be ridiculous.
The most common/most sold usage is a cigarette-shaped system fed with cartridges, which when heated by a coil simulate the smoking of a tobacco cigarette. The "juice" involved can contain anything from 0mg nicotine to 18mg/mL. Of course, any other substance that can be smoked can also be vaped: there are THC vaping juices in places where that is legal, there are caffeine-laced juices for a morning pick-me-up, and some have tried to come up with herbal preparations designed to do anything from enhance lung strength to promote decent sleep. In many places CBD oil, a legal cannabis derivative, is vaped for health and calming benefits, but the vape is not in any way psychoactive.
Because glycols are sweet, the typical e-juice tries to work with sweet flavors like melon, berry, or some variant of candy. Vapor itself can have a thick and flossy mouthfeel, leading to really good attempts at a vanilla custard, butterscotch or other shake like experience. That being said, there are any of a number of companies producing any of a number of flavors, but it appears that your mileage may vary in terms of how you perceive the taste. For this reason, many vaping stores have a "bar" of sorts where, with a disposable clearomizer cover, you can "taste" away at the different flavors to find one you like. Because the flavors tend to the sweet or the dessertlike, there are concerns that this was some deliberate marketing effort to get children involved in addictive behaviors as a gateway to smoking, but in theory you could be vaping something completely nonaddictive, just for the sensory enjoyment of "drinking in" a smokelike substance with any of a number of flavors.
In fact, smoking cessation is one reason why these things have gained somewhat in popularity, as well as their tendency not to permanently stink out anywhere they're used. Because vaporization doesn't produce the tars and other products of oxidation, they're far easier on the lungs, and the amount of nicotine is controllable, it's a safer gateway to tapering off. Some have found that by vaping 18mg/mL, then slowly cutting back the nicotine levels to zero, they manage to acclamate themselves to simply performing vaping as a habit, which is far easier to stop.
This is not in any way to suggest that vaping is a safe habit, which is why it's off limits in most jurisdictions to anyone under 18. And many places, including insurance companies, will treat vaping exactly like tobacco usage - though vaping 0mg/mL is usually not an issue for them. There were some concerns with earlier technologies that the glycerol base could produce aldehydes and other dangerous substances, but these concerns were later determined to be overblown. However, we're not quite sure of the long term effects of vaping as a habit, though what we know right now is that it's infinitely preferable to inhaling burning leaves.
Most typically start with an e-Cig or wandlike apparatus, which is typically charged by screwing the battery end into a threaded adapter that fits a custom USB based charger. That produces enough of a smoke with a cotton wick clearomizer that people can typically try it out for about $20 or so, not counting the juice.
The next level of Rake's Progress is a "box mod", where you purchase a dedicated tank, and a form factor not unlike a large refillable cigarette lighter or box of playing cards. With these units you get the option of replacing the battery, but this also requires a separate charger. The units can typically push more voltage, and last longer between charges - as well as giving precise control of electricity to the coil, controlling temperature and voltage. For your humble narrator's purposes, this can also be used to drive what are known as sub-Ohm tanks, ones whose coils typically operate at less than one ohm. The practical upshot is that you can produce copious amounts of vapor with the push of a button, enabling you to plume a dragonlike spray of thick smoke in any direction, or perform tricks with the vapor that put smoke rings to shame. With this kind of setup, however, most find they really have to ratchet back the levels of nicotine/caffeine/THC/whatever they're used to, because the lung burn or dosage increases significantly.
And for someone who used to smoke and hated being outside, the ability to breathe non-burning, 0mg smoke through his nostrils and/or blow smoke rings through other smoke rings is simply cool, especially when the smoke smells and tastes like cake batter. And frankly, if you're stuffed up in any way, hit a sub-Ohm tank with something heavy on the menthols, and breathe that out through your nostrils for a sinus-clearing experience like no other.
The next level is one where you ditch the pre-built and actually make your own coils, building your own setup from raw wire. It's become a thing at some vape shops to hold tutorials and some guys walk around with tackle-box like kits ready to wire up what amounts to a high-voltage coil. I'm not there yet, but I've seen some people drip a bit of oil onto a custom coil, hit a switch and look like a badly tuned Diesel engine.
Are there risks? Absolutely. Make sure when you take one apart you get the right thread to turn, so that you don't inadvertently open up your tank rather than unscrewing it from the base. Glycols running on to batteries is not good, and inadvertently shorting out your coil has caused more than one person to seriously burn a leg or chest muscle unfortunate enough to be adjacent to the pocket the unit was in. For this reason most people turn their machines off when not actively in use: the industry standard is five rapid clicks to turn on, and five rapid clicks to turn off. Testing on and/or off with a cautious press of the button is advised, and not hearing the characteristic crackling of vaporization is a good sign.
This might turn out, like men playing poker at home, American Choppers and so forth to be yet another silly fad that dies out just after it starts, but for what it's worth the hobby's quite a bit of fun, and there's some possible antibiotic effect from the glycol. Either that or I'll be writing to you from a palliative care ward warning the next generation "don't make our mistake, but in hindsight, we just didn't know." But in the meantime, I think I'll load this up again with that delightful melon daquiri flavor. And deal with the scowl of the cat who knows a plume of vapor is going to be blown above his head.