The modern hero in literature is usually more of an everyman than a classical hero. Modern heroes are complex characters who usually have flaws and problems to which people can relate. They usually don't go on a physical quest; their quest is more of an internal one.
The modern hero, in comparison to a classical hero, is more an "average Joe" dealing with problems that everyone encounters. These may be personal or internal conflicts, philosophical quests for knowledge or self-discovery, journeys toward self-fulfillment or happiness or even quests involving the acceptance of a family member.
Modern heroes also generally do not have any type of special powers or abilities, again making them more like an average person. The modern hero is not the god-like, extra-strong superhero like Hercules, Beowulf or Odysseus. The modern hero is usually a normal person with realistic problems.
Examples of the modern hero include Jay Gatsby in "The Great Gatsby" and Elizabeth Bennet in "Pride and Prejudice." These heroes are both people without any extra power, other than monetary wealth, and they pursue quests that are conceptual in nature. Rather than trying to reach a special place or find a sacred item, they are questing for self-fulfillment.