There is all kinds of cool stuff out there. Last year I was looking at power bricks with wireless charging–some of them combine that plus solar panels and a flashlight. (Of course the solar panels would take forever to charge, but if they are “free energy” while the brick is sitting around, why not?)
My wireless charger is a little finicky - if I have the phone misaligned, it sometimes doesn’t charge. But it is convenient not to have to unplug it if the phone rings (or to wear out the USB port)
iPhones and the AirPods charging case use the Qi charging standard, which is pretty much the de facto standard for wireless charging.
In September 2017 Apple announced it was working on its own wireless charging mat called AirPower which it expected to come to market in 2018. It could do a bunch of things no other charging mat could do at the time, but by March 2019 Apple announced it had canceled the project, ostensibly because they weren’t able to overcome overheating and interference issues.
Such is the blessing of industry standards. There are thousands (millions?) of industry standards out there, including the one that lays out specs for wireless charging technology. No manufacturer has to use it - anyone is free to develop and use their own standard - but instead of their standard being widely adopted, they may find that they’re the only one that uses it.
From that Wikipedia page I found a link to a competing wireless charging standard called WiPower, which sounds vaguely racist; they probably could have chosen a better name.
The station next to my office in London was recently renovated, and they put in smart new wooden benches with little round tables on the armrests. I noticed recently that the tables have a little charging symbol engraved in the wood, and sure enough, if I put my phone on the table, it charges. So now I can sit there, drink my coffee, and add a few % to my phone battery while inviting passing pickpockets to snatch it off the table 
I’m pretty sure my new electric toothbrush I got for Christmas uses this, too, as there was no metal contact that I could find. I suspect that it prevents any risk of shock from the water.
If your electric toothbrush is an OralB model, then yes, it uses inductive charging. I don’t know about other brands.
How does the efficiency of induction charging compare to corded charging? Does it take more electricity to charge a phone through induction rather than plugging it directly to a charger?
It’s 10%-15% less efficient than a cord (early versions were worse). For such low-power devices the extra electricity cost doesn’t tend to bother folks, but you wouldn’t want to use this technology to charge an electric vehicle.
You really, really don’t have to worry about how much electricity it takes to charge a cell phone.
And you need to worry even less about the “vampiric loss” from leaving the charging brick plugged in when you’re not charging your phone.
For cordless charging, is the efficiency of charging affected by the orientation of the phone on the pad? I would guess not, at least in terms of its orientation in the plane of the pad.
I am less sure I understand that would still be the case if the phone (and the coils/battery within it) were tilted at an angle (up or down) to the pad.
Does the orientation matter?
Angular orientation (i.e. whether the ear-end of your phone faces N/S/E/W) shouldn’t matter. But the coil inside your phone needs to have its axis aligned relatively well with the axis of the coil in the base. This assures that the maximum amount of magnetic flux produced by the base’s coil passes through the coil inside the phone. Part of the Qi wireless charging standard includes communication from your phone to the base telling it how much power is being received; the base adjusts its coil current as needed to achieve the target power level in the phone’s coil. Presumably the base has an upper limit on how much current it will push through its own coil; if your phone is positioned badly enough so that the base hits this current limit while trying to deliver the target amount of power, it will shut down (don’t know for sure, but this might even be how it knows to shut down when you simply pick your phone up).
That may be why the charger I have is designed the way it is. There’s not much doubt how to align it, standing up against the vertical piece.