Offspring
The offspring are strictly speaking the progeny, or young, born to a person.
In a larger sense, the offspring are collectively all of the descendants, brood or family.
For example, the offspring of someone with a genetic (inherited) condition such as Huntington disease are themselves at risk for the disease.
'Offspring' springs from the Middle English 'ofspring' which sprang from the Old English 'of' meaning 'from' + 'springan' meaning 'to spring' = to spring from.