Despair is the feeling of not having any hope left. If you completely forgot to study for your final exam in math, you might feel despair when your teacher passes out the test.
Despair can also refer to someone or something that causes you to worry or be sad. Unfortunately, some children are the despair of their parents. The verb despair means to lose hope. Despair is from Latin desperare "to be without hope," from the prefix de- "without" plus sperare "to hope," from spes "hope."