verb

• (intransitive) To inhale a larger quantity of air than usual, and immediately expel it; to make a deep single audible respiration, especially as the result or involuntary expression of fatigue, exhaustion, grief, sorrow, frustration, or the like.
  "When she saw it wasn't damaged, she sighed with relief."
• (intransitive) To lament; to grieve.
  "He sighed deeply in his spirit."
• (transitive) To utter sighs over; to lament or mourn over.
  "Ages to come, and men unborn, / Shall bless her name, and sigh her fate."
• (intransitive) To make a sound like sighing.
  "And the coming wind did roar more loud, / And the sails did sigh like sedge."

noun

• (figurative) a manifestation of grief; a lament.
• (Cockney rhyming slang) A person who is bored.