Does anyone know the names for each relative in mandarin?
This may help:  親屬稱謂

Notice:
爺爺 = 祖父
奶奶 = 祖母
外公 = 外祖父
外婆 = 外祖母
伯伯 = 伯父
叔叔 = 叔父
姑姑 = 姑母
嬸嬸 = 叔母
舅舅 = 舅父
舅媽 = 舅母
爸爸 = 父親
媽媽 = 母親
阿姨 = 姨母
堂哥 = 堂兄
表兄弟姊妹 from 姑姑's side are 姑表
表兄弟姊妹 from 姨母's side are 姨表
Husband and wife are 丈夫 and 妻子 (also:老公 and 老婆)
Great grandfather and Great grandmother are 曾祖父 and 曾祖母; 外曾祖父 and 外曾祖母
your 妻子's 哥哥 is your 大舅
your 妻子's 弟弟 is your 小舅
your 妻子's 姊姊 is your 大姨
your 妻子's 妹妹 is your 小姨
Your 丈夫's 哥哥 is your 大伯
Your 丈夫's 弟弟 is your 小叔
Your 丈夫's 姊姊 is your 大姑
Your 丈夫's 妹妹 is your 小姑