Swarm behavior is in many ways the quintessential example of emergence. Individuals in a group act accordingly to those around them which causes the entire group of them to move of behave as a large unit.
Such behavior can be seen in many parts of nature including flocks of birds, schools of fish, and ant colonies.
To see an example of bird flocking go here.
Swarm behavior can also be found in certain human interactions. One potent example is in mass protests and marches. Of course, this interaction comes in a ruly and unruly version, both being equally emergent and are simply founded on different basic interactions. The unruly variety can be likened to a flock of birds where the basic interaction is fueled by cooperation whereas the ruly variety has a basic interaction fueled by anger or discontent.