A character achieves Hero status by True Reincarnating once. This is a 34-point build (30-point for Drow).
After True Reincarnating once and achieving 'Hero Status', you gain 'wings' around your character's name, which are visible to other players. Your character will also be very slightly taller than an adventurer or champion.
Hero character on non-VIP account (Premium/Free to Play) can only open quests on Casual, Normal, Hard to begin with. Elite and Reaper must be unlocked via completion or store purchase.