The short story: this is a Cockcroft-Walton generator hanging off a resonant transformer. If you don't mind wasting a couple minutes with detailed theory then charge ahead intrepid reader! Otherwise skip to the next step.
The long story, well, it's not much longer. Take a coil, make it resonate at a particular frequency using a capacitor, then place it near a similarly tuned coil and use the oscillating magnetic field of the first to cause the second to resonate. Use a clever AC to DC converter and voila, you have a method of wireless energy transfer.
After some sleuthing on the internet, I went about devising the first part, an oscillator. Various homebrew methods have been used (see: Wireless Power Instructable) but weren't very good or just temporary solutions. I used the suggestion on wikipedia of using a Colpitts oscillator. This is a decent solution because it's dead simple to build and, most importantly, it's a current oscillator and not a voltage oscillator. As current through an inductor is what generates the magnetic field, this is what will drive both coils.
The second part is fairly easy to understand, that being the two coils. Although they don't have to be the same physical size, they do need to resonate at the same frequency. The combination of number of turns and diameter determine the inductance, and some capacitors were added to obtain the correct oscillating frequency. It gets tricky when you get into the details however (and they get very, very detailed, so I won't put the majority down here) as you need to select the diameter of wire to go with the amount of current going through your coil, which will determine the amount of resistance in the coil, which will impact the viability of your oscillator. To make it somewhat easy, go with 24AWG enamled magnet wire.
You now get to pick a some-what arbitrary frequency for your circuit. This I decided to go with 80KHz, it happened to be a nice middle ground between easiness and efficiency. Then you pick a capacitor value that's commonly available, I picked 150nF. This took a while to select because you need to get an inductance that is within the realm of being hand made. Using the equation:
frequency = 1/( 2 * pi * sqrt(inductance * capacitance / 2) ) (from Colpitts oscillator)
we use the capacitor and frequency values to try to get the inductance in and around 20uH to 70uH. Air-core inductors around those values are easy to make. I used a value of 53uH, which to calculate:
freq = 80k
cap = 150nF
freq = 1/(2*pi* sqrt(ind * cap/2) )
freq * 2 * pi * sqrt(ind * cap/2) = 1
ind * cap/2 = (1/ (freq * 2 * pi) )^2
ind = (1/ (freq * 2 * pi) )^2 / (cap/2)
ind = (1/ (80k * 2 * pi) )^2 / (150nF/2)
ind = 52.77uH
From here you need to use this handy inductor calculator to try to figure out what diameter and number of turns are needed. I used values of ~22 turns at 6cm diameter, with an arbitrary length around 4-5x the wire thickness for the secondary, and ~13 turns at ~15cm diameter for the primary. These values will be your STARTING POINT ONLY. You have to experiment to get it right (covered in the next couple steps).
Note that you are using the same inductance and capacitance for both the resonating coils, this is so it's easy to tune. Don't go crazy with different inductances and capacitances or else you won't get it to work.
OK, the last part of this picture is the AC to DC converter. This is what will shape the received AC into something we can use to charge a capacitor or a battery at a usable voltage. I used a CW generator here to great effect; it allowed me to tune the slave coil to produce exactly the right voltage without going over the charging voltage. I determined (through experimentation) that a two stage generator would be enough, and that will generally be fine when trying to generate ~5V. For the capacitors I arbitrarily chose 2.2uF caps, and for the diodes I chose a nice Schottky diode array with a very low 0.38V forward voltage drop. The P/N is BAS40TW-TP, however these are VERY small parts so you will probably have to order individual schottky diodes for this one. Just use ones with a low voltage drop AND a low reverse leakage current.
OK! Enough of this long-winded theory and background info, let's get to the actual good stuff!