Examples of hopes include hoping to get rich, hoping for someone to be cured of a disease, hoping to be done with a term paper, or hoping that a person has reciprocal feelings of love.
Hope was personified in Greek mythology as Elpis. When Pandora opened Pandora's Box, she let out all the evils except one: Hope. Apparently the Greeks considered Hope to be as dangerous as all the world's evils. But without hope to accompany all their troubles, humanity was filled with despair. It was a great relief when Pandora revisited her box and let out hope as well. It may be worthy to note that in the story, Hope is represented as weakly leaving the box but is in effect far more potent than any of the major evils.