One example of the influence of postmodernism on people's thought would be the rise of cultural relativism. Postmodernists tend to question whether there is a single set of assumptions or norms that all people should live by. So a person who views different cultural systems or beliefs as being more or less equal is embracing a view associated with postmodernism. Another postmodern perspective might be someone who questions what we are capable of knowing, even through such accepted disciplines as science or history. One example of this might be a historian who questions how we can construct narratives to explain events in the past.

How does Postmodernism influence people's behavior?

I think that the most common element of people's thought or behavior that results from Postmodern thought is the idea that "truth" is multifold and extremely complex.  The Postmodern remnant on intellectual thought is the idea that the Platonic notion of "T"ruth is not evident.  Rather, there are different valences of truth.  There is a proliferation of information and technology designed to understand constructions of truth.  These help individual gain more comprehension, but little in way of absolute understanding.  The lingering element of individual thought and behavior arising from Postmodernism is this idea that the search for truth is manifold and something that is not immediately direct in its pursuit.  This disintegration of that which was seen as absolute is one of the legacies of Postmodernism.  It is evident in how individuals appropriate understanding based on the presence of the world wide web and information technology.  There is no longer simply "one source" of understanding.  Rather, there are many sources and many avenues to pursue.  Access is more, information is more, but overall understanding might be more elusive, infinite, almost.  In this, one sees where Postmodern thought is present in people's thought process.