Socially speaking, a class is different from a caste
in that an individual's class may change throughout their life
or from one generation to the next.
Most Western countries have a class system, to one degree or another. If your parents are poor you will probably be poor too, but you might improve your situation through hard work or luck. Likewise, you might be better able to provide for your children.
In a caste system, the social group you are born into is the caste you will die in, period. I believe India still has a caste system, which is closely tied to their religion. They believe that if you do well in life, then in the next life, you may be reincarnated into a more prestigious caste. This belief helps provide stability to a social system that would otherwise be weakened by the great number of people in low castes who are helpless to do anything about it within their culture.