Dont replaces de + a noun inside a relative clause, most often corresponding to "whose", "of which", or "about which": c'est l'ami dont je t'ai parlé (that's the friend I told you about — literally "about whom I told you"). It's required whenever the verb or expression in the clause takes de: parler de, avoir besoin de, être content de.
Dont is also the standard way to express "whose" in French: c'est la fille dont le père est médecin (that's the girl whose father is a doctor). Word order after dont stays as subject + verb + object, unlike English "whose", which pulls the possessed noun to the front.