Why is volcanic ash hazardous?
As mentioned in the comments BA flight 9 is a thorough example of why ash can be a huge issue.
Relevant points of that incident
One issue that ash creates is it cannot be seen on radar so it can be hard to track at night and must be reported by other means. This is deceiving as you would physically see what appears to be a cloud yet weather radar would not show anything.
Inside the cabin, this depends on the granularity of the ash. Some of it will be filtered by the plane's air filters as any particulate matter would. However, ash can be extremely fine and what does make its way through the filters will present itself in a smoke like manner. This may cause alarm in the cabin and a false indication of a fire.
When it comes to the exterior of the plane there are a few issues. The fine grains hitting the plane at that speed will act like sandpaper. The windshield may become scratched to the point you can't see through it (which is not actually as big of an issue as some may assume). The paint will be effectively sanded off which in the short term is not a huge issue. More importantly it could affect your instrument inputs by clogging things like the static ports and pitot tube creating erroneous readings in the cockpit.
The engines were the main issue that arose on BA 9. In short the ash entered the engine, was melted and began to adhere to the blades of the jet. In turn this blocked the engine airflow and caused an engine stall. In the process BA 9 lost all four engines; however, when they began to descend the solidified ash cooled and broke off the engine allowing a restart.
Air crash investigation covered the BA incident nicely in season 4. You can watch it here.
Other risks that BA 9 did not have
For a piston plane ash could potentially clog the air filter and starve the engine of air. Any super-fine ash that could get through the filter runs the risk of being ignited and causing detonation issue with in the engine. Likewise the sanding/grinding issue could have adverse effects on the propeller shape reducing its efficiency or even overall ability to act as an airfoil.
There is a nice wiki on it here and it appears that as of 1991 centers called &quot;Volcanic Ash Advisory Center&quot; have been set up to help track these very issues.