Literary conflict defines the obstacles the protagonist or protagonists must overcome in order to succeed. Their struggle against these obstacles forms the action of the story, which rises until the climax at the end of the story. The hero then resolves the conflict either by triumphing against his obstacles, or falling to them.
Internal conflict means conflict within a given character: struggling against his fears, his doubts, his moral failings and similar emotional or intellectual conditions. External conflict entails a conflict with forces outside the hero's own mind: fellow men, the natural world, or larger societal organizations. Of the four basic types of conflict, only one--man vs. himself--is internal. However, a well-written novel may contain multiple types of conflict, blending external and internal.
The most basic type of external conflict is man vs. man. It pits the hero against another person or people bent on defeating him. For instance, Shakespeare's play "Henry V" shows King Henry battling against the French nobles for control of their country. Not every instance of man vs. man conflict need be overt however. Sherlock Holmes, for example, often pits his wits against that of a criminal mastermind: a duel of intellect and will rather than a physical fight.
Man vs. nature encompasses a human being's struggle against the weather, the wilderness or some manner of wild beast. It can also entail a conflict again more nebulous natural forces, such as the struggle to find a cure for a disease, or the fight to get crops to grow. Jack London's stories often focused on man vs. nature conflicts, as does the novel "Robinson Crusoe."
A man vs. society conflict depicts an individual struggling against larger social forces, such as governments, cultures or societal trends. Though the conflict entails nebulous concepts such as bigotry or government hypocrisy, it's still consider external conflict, since the struggle takes place outside of the protagonist's mind. Examples include "1984," in which Winston Smith tries to keep his soul against an oppressive government regime bent on destroying it, and "To Kill a Mockingbird," in which lawyer Atticus Finch attempts fight the injustice of racism in the deep South.