They controlled the life and destiny of everyone. Clotho spins the thread of life (begins a person's or creature's life), Lachesis measures it (looks at the how long it currently is), and Atropos cuts the thread. When the thread is cut the person dies. The Moirai are capable of destroying an immortal.
The decisions of the Moriae about a person's life cannot be changed. Even Zeus is powerless to change their will.
Their Roman equivalent were the Parcae.
Sources: D'Aulaire's book of Greek Myths
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Moirai Facts for Kids. Kiddle Encyclopedia.