One of the first films with a narrative storyline was The Great Train Robbery, which premiered in 1903. Even then, the conflicts of cinema essentially broke down into four different categories. The ancient Greeks established these categories of conflict in their plays and while there variations and hybrids exist, they continue to be the basis for the bulk of films you see today. Every movie in one way or another has a conflict.
"Man Against Himself"
"Man against himself" is one type of conflict you'll find in many films. "Man against himself" is when your lead character's biggest obstacle is himself. One example would be 2002's Adaptation, which tells the story of its main character overcoming his low perception of himself. Walk the Line, which came out in 2005, tells the story of Johnny Cash as he tries to overcome his own dependencies on drugs.
"Man Against Others"
"Man against others" is when your protagonist faces external conflict from the people around him. 2001's The Royal Tenenbaums is loaded with this. It tells the story of a dysfunctional family battling each other and laboring to find a way to coexist. Sports movies usually contain solid examples of "man against others" conflicts. Rocky IV is about Rocky Balboa physically and emotionally preparing to box the giant Ivan Drago. The entire Rocky series gives a different example of this conflict.
"Man Against Society"
When your protagonist must face society as a whole it is called "man against society." You'll find a lot of political films like 2008's Milk that feature this conflict. Milk tells the story of Harvey Milk, the first openly gay elected official in the state of California, as he works to establish laws that protect and further the civil rights of homosexuals. You can also find "man against society" conflicts in science fiction films like Logan's Run. Logan's Run is set in a future when a person must submit to execution at the age of 30. The story of this movie revolves around Logan as he tries to escape his fate.
"Man Against Fate or the Universe"
"Man against fate or the universe" is also known as a situational conflict; situational in that it puts your seemingly normal protagonist in an extraordinary situation. Jurassic Park is a great example of this. Jurassic Park finds its main characters stuck on an island that is populated by dinosaurs. Hollywood Ending is a Woody Allen film that has a situational conflict. In it the protagonist, a film director named Val, goes blind but must continue to direct his latest film.