Information depth in x-ray photoelectron spectroscopy (XPS)
Do you understand the basics of XPS?  The Wikipedia article is pretty clear. In summary, a (usually monochromatic) x-ray source irradiates a sample.  The x-ray can be absorbed by electrons in the sample, and can provide enough energy for a core electron to be set free with an energy equal to the incident x-ray energy minus its binding energy. Collecting the electrons and measuring their energy gives a plot of counts vs electron energy.  Lower electron energy indicates an electron coming from deeper in the atomic shell. Given knowledge of atomic binding levels, you can associate specific peaks in the spectrum with elements (with chemical effects on binding energies giving additional information). 

So - what is the "information depth"? A slightly odd term, I've generally heard/used 'detection depth'.  Two questions to ponder - (1) how far will x-rays of a few keV penetrate into a solid (if an atom doesn't see any x-rays, it won't emit any electrons), and (2) how far can a ~100eV-1000eV electron travel in a solid before being scattered (and thus losing the useful info).  Not surprisingly, (2) dominates for XPS, and is often called the 'escape depth'. 

As for the 'take-off angle' - again, the interesting part is where your electron detector is placed relative to the sample, and how far the electrons have to travel through the sample to head out in the direction of the detector - the 'escape depth' will vary with exit angle (take-off angle) since the amount of material the electron has to go through is a simple trig function of the exit angle.

I hope that helps.