There are very different approaches depending on which country or region you're talking about. In general, though, aristocratic children who weren't about to succeed their father's lands and titles would either:
Pursue a military career, usually as a knight, serving under either their father, older sibling, sovereign, or under a knightly order like the Templars.
Join the clergy/church.
Be given administrative responsibilities and aid their older sibling in an supporting role.
Be granted a lesser title and smaller fiefs of their own and become part of a cadet branch of the main family.
Do absolutely nothing, whiling away their days for as long as they could count on the goodwill of their father or their older sibling.
I could give you a more specific answer if you'd like to specify an actual country or region?