If asked about conflict, I would describe myself as conflict-adverse. I would prefer to avoid it, but I will step up and fight if all other avenues have been exhausted. This stance makes writing about conflict extra tricky for me. Life is full of conflict (internal, external), and avoiding it in fiction is unrealistic (this coming from the person who writes about vampires, wizards, and fairies).
Character versus self conflict includes a moral dilemma or a mental health condition. It demonstrates a struggle the character has within her/himself. The conflict resolves when the character makes, and acts on, a choice. Fight Club by Chuck Palahniuk is a great example of this type of conflict as demonstrated with the main character's interactions with his alter ego.
Character versus character conflict occurs when the needs and/or wants of one character contradict those of another character. The characters are on opposing sides whether it's a straightforward fight or a complex power struggle. The conflict resolves when the characters needs and/or wants align or when one character defeats the other. In William Goldman's The Princess Bride, the hero Westley must defeat the prince engaged to Westley's true love Buttercup.
Character versus nature conflict is present when the heroine/hero must survive against poor weather, harsh environment, or natural disaster. The conflict resolves when the protagonist survives or dies. There is always a possibility that nature wins. For example, Into the Wild by Jon Krakauer, a story about a young man who decides to live off the land in the Alaskan wilderness and dies.
Character versus supernatural conflict occurs when the character must battle a supernatural element, like ghosts, vampires, or other mythical creatures. The supernatural characteristics create a power imbalance which makes it even more impressive if the hero prevails. Often both parties possess supernatural skills, and throughout the story arc, the hero learns how to hone his/her power. The Percy Jackson and the Olympians series by Rick Riordan immediately comes to mind as I picture the epic battles between the young demigod and the monsters he encounters throughout the books. As the books progress, he gains confidence and experience in his skills.
Character versus society conflict is present when the character opposes what is considered "normal" for the purposes of, for example but not limited to, survival, morality, or love. In George Orwell's 1984, Winston questioned the Party in his diary which was a crime ("thoughtcrime") punishable by death. He wanted the freedom to think which threatened the status quo of a tightly controlled society built and maintained on propaganda and outright lies.
Conflict is essential to drive a narrative forward. It is the catalyst for character growth. And it keeps life interesting.