A gift given to relatives or close friends on the occasion of a family celebration (often as a return gift for gifts received at events like births or weddings).
出産の内祝いにタオルセットを贈った。
I gave a set of towels as a gift for the birth celebration.
結婚の内祝いは親戚だけに贈ることが多い。
Wedding return gifts are often given only to relatives.
内祝いには熨斗（のし）を付けて渡すのが一般的だ。
It's common to attach a noshi (ceremonial wrapping) when giving an uchiiwai.
A private family celebration or private ceremony for relatives (a celebration held only among close family).
今回は内祝いとして親族だけで食事会を開くことになった。
This time we decided to hold a family-only meal as a private celebration.
大きな式はしないで、内祝いに留める予定だ。
We won't have a large ceremony; we plan to keep it as a private family celebration.