This is a huge question and an answer here can only begin to scratch the surface. But let's give it a shot. First, postmoderns question the ability of language to convey anything more than a partial, incomplete truth. While in the 18th century, during the Enlightenment, thinkers largely believed that writing in the most accurate possible language would provide a clear windowpane into truth, postmoderns question that assumption. They tend to stake their claims on Nietzsche's contention that language is a "prison house." What Nietzsche meant was that we are born into a language that carries with it limitations and stereotypes, and as we learn the language, we internalize its limits. One example of this would be Harriet Beecher Stowe's Uncle Tom's Cabin. She wrote the book with the fullest sympathy for black slaves and made a passionate argument for freeing them, but at the same time was caught in a "prison house of language" such that she could not get beyond the racist stereotypes of her culture. So postmoderns ask, what are the limits of language? What is being left unsaid? What are we assuming to be true that might be false (such as Stowe's belief that blacks were "childlike")?
Second, in the same vein, postmoderns would argue that we can't "know" without acknowledging or being conscious of our "social location." In other words, they argue that there is no place--no mystical mountaintop-- where we can position ourselves to see all truth. While there might be an ultimate truth, no human can see it, because we are all blinded in one way or another: a woman cannot see the world the way a man does, a person of privilege can not see the world as an oppressed person does and a white person can't see the world as a black person does, at least not entirely. Whole cultures have blind spots too. So postmoderns reject the idea of a transcendent knowing. A first question they might ask would be, "what is the social location of the person writing this book or essay?" so that they can understand where the writer is coming from. (A way to understand this would be to go to Youtube and watch a version of the "Blind Men and the Elephant.")
Since postmoderns tend to want to contextualize knowledge in history, it might be useful to quickly rehearse what has happened in recent history. While postmodernism arose in the 1960s with the ascent of thinkers such as Jacques Derrida and Michel Foucault, it came, in part, out of the profound questions raised in Western culture at the end of World War II. (Bronowski's Ascent of Man's episode on the concentration camps offers insights into this.) How did all our knowledge and so-called cultural and intellectual superiority leave us standing in the ruins wrought by genocide, the atom bomb and the kind of widespread destruction unleashed by totalitarian ideologies? What didn't we see? Where did we go wrong? Where might we still be going wrong? What might the people once dismissed as "savages" have to teach us? Postmodernism, much as it has been maligned, arguably brings a profound humility to the intellectual table.