Used item of value given to someone else. It refers to someone wearing a new shirt, for example, and after a  year or so, they hand it down to someone else to use. Hand-me-downs are common in families with many children, as kids grow out of shoes or clothes, they are handed-down to the next  younger child. It may also imply a family heirloom : “This ring I am giving you is a hand-me-down from my mother.”

“I was the youngest of 4 brothers, so I just got hand-me-down clothes, never anything new.”