To desensitize is to make less sensitive, or less easily upset. Some people say that watching scary movies desensitizes kids to violence.
A shot of novocaine desensitizes you to the pain of having a cavity drilled by your dentist, and lots of spicy food can desensitize you to more subtle flavors. In both cases, you become less perceptive, or less affected by a sensation or taste. Witnessing violence can likewise desensitize soldiers or war reporters. The word originally referred to photographic development, from de-, "do the opposite of," and sensitize, with its Latin root sentire, "feel or perceive."