In apparel, private label ranges from relabeling a ready-made blank with your own neck label and hang tag to fully custom cut and sew. Either way, the finished product carries your branding and no visible trace of the original maker.
Relabeling usually means removing the original tag, which is easiest with tear-away labels, and adding a printed or woven label. In the US, textile products sold to consumers generally need fiber content, country of origin and the identity of the maker or seller on a label, plus care instructions, so the new label has to carry that information correctly.
Ask what finishing is included (label removal, printing or sewing in new labels, tags, bagging) and who supplies the label content. The private label finishing guide walks through each piece.