Meaning:

A shirt, dress, pants, etc. that an older person (especially your older brother or sister) in your family has given to you because it is too small or he/she doesn't want to wear it anymore.

A hand-me-down is a piece of clothing that has been worn by someone (especially your older brother/sister) and given to you.

Example:

When I was a child, I always had to wear my sister's hand-me-downs.

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