A black hole has an event horizon which 'marks the point of no return'. So yes, light cannot escape from a black hole.
Why? Well, think of a 'spacetime fabric'. It's the easiest way to understand the physics at work here, in my opinion.
Usually, the fabric would look like this:
fabric http://whyfiles.org/wp-content/uploads/2011/05/gravity_probespacetime.jpg
(source: whyfiles.org)
However, a black hole has so much gravity that one could say it 'rips' the spacetime fabric:
(source: ddmcdn.com)
When the light hits this area of amazingly intense gravity, it simply cannot get out - the light travels 'along' the fabric, and since there is a rip in the fabric, one could say it simply goes away - it becomes part of the singularity.
This is a simplification, of course, but it's enough to understand at least part of the physics behind this phenonenom.