The suffix '-ides' in Ancient Greek
means 'son of' and this phrase means simple: 'the son of Atreus
Atreus was a mythic Greek king, the son of Pelops, who fought with his brother Thyestes and served him a dish made of 49 of his sons (Thyestes' sons that is).
When used in the sigular, this is one of the names of Agamemnon, and he is often referred to as 'Atreides' in the Iliad and other poems, plays etc.
When used in the plural, the Atreidai mean Agamemnon as well as his brother, Menelaus.