Processing is an open source language/ development tool for writing programs in other computers. Useful when you want those other computers to "talk" with an Arduino, for instance to display or save some data collected by the Arduino.
Arduino comes with some basic examples for communicating with Processing (in Examples > Communication). These are useful for when you want to write both Arduino and Processing programs and have them talk to each other. This works best for communicating simple information.
know more: Arduino Playground - Arduino and Processing