You hear many different answers: 1, 3, 7, 20, etc. The fact is, you can divide the types of stories down in so many different ways that it doesn't really matter.
Good stories are about conflicts. Here are some of the more common conflicts you might see in writing:
- Human vs Nature - people combat a force of nature such as a fire or flood
- Human vs Self - people combat their own desires and weaknesses
- Human vs Human - hero vs villain
- Human vs the Environment - people combat the environment, such as survival in the wilderness or colonizing a new area
- Human vs Machines/Technology - people combat technology
- Human vs the Supernatural - people combat something supernatural, such as a vampire or demon
- Human vs Society - people combat the rules of their society
Ronald Tobias published a book where he suggested that there are 20 Master Plots; and Georges Polti suggests 36 Dramatic Situations - see Link below for lists.