To exercise is to engage in activity meant to improve one’s physical fitness (and of course the word has many other definitions, none having to do with casting out spirits). To exorcise is to purge something spiritually bad. Generally, what is exorcised is the evil entity that is cast out, but the word sometimes takes as its direct object the person from whom the evil entity is cast out (as in the first and last examples below).
Most English speakers are comfortable with exercise, so here are a few examples of exorcise put to use:
Thus all converts from Paganism and Judaism, and even the children of Christian parents were exorcised before being baptized. [Nicene and Post-Nicene Fathers, Philip Schaff]
Despite being depleted by injury, South Korea are keen to exorcise the demons of the 2009 final, decided after extra innings. [Eurosport.com]
We’re not advising that you attempt to exorcisedemons or perform hocus pocus, but like a good spring cleaning, exercise can banish the cobweb. [Overcoming Anxiety for Dummies]
Twenty-five years later, at a low ebb, he began writing the script that would exorcise the memories. [Sydney Morning Herald]
They find out the woman the priest was exorcising died after having a seizure. [TV Equals]