姐 = 女 (woman) + 且. The woman radical marks it as female; 且 lends the sound. It means an elder sister.
姐姐 (jiějie) is an OLDER sister (younger = 妹妹). Beyond family, 小姐 (xiǎojiě, 'little miss') is a polite way to address a young woman — though in mainland China it's now used carefully, as it can carry other connotations in some contexts; 女士 (nǚshì, 'madam') is safer formally. 大姐 respectfully addresses an older woman.
A woman (女) — your older SISTER.
姐 + 妹 makes 姐妹 (sisters); 兄 + 弟 makes 兄弟 (brothers). Pair an older and younger term and you get the whole category.