Nonviolence is a concept in which the fundamental belief is you do not need to resort to violence in order to achieve your goals. This includes hurting other people, animals or even the environment, and usually it's used as a method of political protest.
Perhaps the most famous nonviolent protestor of all is Mahatma Gandhi, a leading figure in the Indian nationalist movement. His nonviolent protests and willingness to stand up to British rule brought the cause of Indian independence to the forefront of the social consciousness in both India and Britain. He proved that to achieve political goals, you don't need to resort to fighting.
A couple of quotes from Gandhi best sum up the concept of nonviolence:
"An eye for an eye makes the whole world blind."
"There are many causes that I am prepared to die for, but no causes that I am prepared to kill for."