Since Black holes are basically a type of gravitational well absorbing matter at varying rates, since it’s taught that matter cannot escape the gravity of a black hole, why then X-rays escape the gravity effects of a black hole?
Good questions. X-rays don't escape from _inside_ the black hole, but rather come from hot material right _next to_ the black hole. Once that hot material goes inside the black hole, no X-rays or anything else ever come out again.
Why does stuff get hot when it falls into a black hole? Basically, as stuff starts falling toward a black hole, it moves faster and faster (just like if you drop something, it falls toward the Earth faster and faster). However, the stuff falling toward the black hole hits _other_ stuff that's also falling toward the black hole. The effect of this is to heat everything up. This is just like what happens when your hand heats up if you rub it back and forth on your desk. Finally, why does hot stuff emit X-rays? Well, we know that hot things glow–if you put an iron rod into a fire, it will eventually glow red, and then white. If something gets _extremely_ hot (several million degrees) it emits not visible light, but X-rays.
So that's the story. Here's a nice artist's impression of the whole thing: http://en.wikipedia.org/wiki/Image:Accretion_disk.jpg
The X-rays are coming from the disk of material around the black hole.
Take care, Greg
Image source: Wikipedia