There are two main types of argument within the area of speech-writing and rhetoric. Many stories use subtle types of argument by making their readers think about an important concept without necessarily bringing about any type of action or change of mind. However, these two are clear types of argument that have clear intentions of influencing the audience in some way.
Aristotelian is an adversarial type of argument. It originates with Aristotle, a well-known Greek philosopher and orator from around 384 B.C. The speaker or writer is attempting to change the minds and actions of the audience. Logic, emotion, and credibility are the three main elements within the text as the author attempts to persuade those listening or reading the argument. Other concepts within the speech would be the invention, arrangement, style, memorization, and delivery.
Rogerian is a consensus-building type of argument. It originates from the psychological work of Carl Rogers. The writer is attempting to create cohesion between the readers to bring them together on a topic. People generally only read literature that appeals to them or follows their beliefs, so narratives, poems, etc. that reinforce these beliefs will be more popular.