Cease, very simply, means stop. It often carries a sense of urgency, for example if one person asks another to cease doing something.
The hailstorm had begun explosively, with large hunks of ice banging on the house's tin roof. And then, just as suddenly the noise ceased as the violent storm moved away from us.
After seventeen hours, and with Dr. Roberts' help, Hassan's hiccups finally ceased. The only problem now was that he started yawning uncontrollably. The doctor is currently searching for a cure.
Cease and desist is a legal term for when an individual or company receives an order from a judge or lawyer to the effect that he (or they) stop what they're doing and refrain from starting again. In military battles, a cease fire is a situation wherein both sides mutually agree to stop fighting. Cease fire can also be used figuratively, as in cases when two arguing parties agree to stop the argument.
Jack's company, Exotic Effects, had to stop production on their bestselling guitar effect, the Freaky Fuzz, when they got a Cease and Desist letter from the lawyer for a company called Snide Sound works, claiming that they had been selling something called the Freaky¬Deaky Fuzz for more than three years.