This type of conflict is the protagonist's struggle with a force outside of himself, like another character, a group of people or a force of nature. It also can be an opposition between themes, ideas and other intangible concepts. An example of external conflict is in Herman Melville's "Moby Dick," where the narrative of the whole book revolves around Captain Ahab's pursuit of the whale.
There are four main categories of external conflict. Man versus man occurs when the protagonist experiences discord against other men. Man versus society indicates a struggle against ideas, practices or customs of another culture. Man versus nature usually applies to stories about a character lost in the remote wilderness or fighting a natural disaster. Finally, man versus machine plots the protagonist against man-made entities. This final category is usually relegated to science fiction stories, such as Isaac Asimov's "I, Robot."
Internal conflict is the struggle within the protagonist. Usually it involves the character making a hard decision, overcoming personal angst, fighting temptation or something similar. An obvious example of internal struggle is the book "Sophie's Choice" by William Styron, where a woman in a concentration camp is forced to choose which of her two children will die. In many cases of internal conflict, the line between right and wrong isn't easy to decipher. Both the character and the reader are often torn between different solutions to the problem.
The major category of internal conflict is man versus himself, where the main character struggles within himself to decipher ideas of right or wrong, physical limitations or other concepts. Shakespeare's "Macbeth" is a prime example of this type of conflict. Macbeth's own ambition is constantly battling against his conscience. Man versus destiny is also considered internal. The protagonist tries to break away from a pre-determined path, or fate, to follow his own free will.