Despair is the familiar word in this pair. As a noun, it means a state of having lost all hope: After months without good news, she fell into despair. It can also be a verb meaning to lose hope: Do not despair; we still have time to solve the problem.
Respair moves in the opposite direction. It is a rare noun meaning renewed hope or recovery from despair. For example: When the missing hikers were found alive, despair gave way to respair. The sentence describes hope returning after a period of fear and hopelessness.
Although respair provides a neat contrast with despair, many readers will not know it. In most everyday, academic, or professional writing, renewed hope, fresh hope, or recovery from despair will communicate the idea more clearly.
- Despair: Hope is lost. He despaired when every attempt failed.
- Respair: Hope returns after being lost. The promising results filled the team with respair.
Choose despair when someone has no hope or is losing it. Use respair only when its rarity suits the style or when the contrast with despair is important. Otherwise, renewed hope is usually the more natural choice.