Does Wi-Fi radiation pass through my body?
Ionizing radiation is a little complicated, so stay with me as I try to explain it in an easy way...

When talking about ionizing radiation, scientists talk about energy levels.  But this refers to the energy level of the photon of electro-magnetic energy, not the quantity of photons.  All electromagnetic energy (radio waves, light, x-rays, etc.) can be thought of as either a wave or a particle.  The shorter the wavelength, the higher the energy.  So when scientists talk about the energy level of ionizing radiation they are talking about the wavelength.  

Here's a wiki page showing the electromagnetic spectrum.

Only the higher energy waves are ionizing.  Specifically, stuff in the UltraViolet and above (X-Rays and Gamma Rays).  Stuff in the visible spectrum and below (including radio waves and microwaves) are non-ionizing.

Wifi signals, which are in the 2.4 to 5.something GHz range are not ionizing.  

I should point out that if something isn't ionizing then simply having more of it (at the same frequency/wavelength) is not going to make it ionizing.  It doesn't work that way.  

Non-ionizing radiation can have an effect on your body, however.  It can cause heating.  A microwave oven, for example, operates near 2.4 GHz and obviously heats up food.  But a microwave does not ionize food.  

But let's put all of this into perspective.  A typical WiFi device can output about 0.1 watts of energy.  A typical LED flashlight will output about 1 watt of light.  They are both non-ionizing energy and will have a similar heating effect.  The main difference is that the flashlight will heat you 10 times faster and in a more concentrated spot on your body.  Yet you wouldn't think twice about shining a flashlight on your hand-- and you shouldn't worry about it.

On the equator at noon the sun puts out approximately 1,000 watts of energy per square meter of ground.  The vast majority of this is non-ionizing (the UV part is ionizing).  This is about 1,000 times more "radiation" than the LED Flashlight, and 10,000 times more than the WiFi signal.  You run more risk going outside than sitting in your house playing on the iPad.  Even so, just put on some sunscreen and enjoy the outdoors!

Some electro-magnetic radiation will pass through your body.  The higher wavelengths and lower wavelengths in particular will pass through more easily.  But passing through means that their energy did not interact with your body.  It's the stuff that doesn't pass through that you're interested in.  Even so, what I said above assumes that 100% of the energy gets trapped in your body and it still isn't an issue.

Conclusion:

A WiFi signal is non-ionizing and is thousands of times less energy than going outside in the sun.  Don't worry about it.  It's not going to harm you.