The Chinese word for grandmother is
祖母

zǔ mǔ

Chinese Definition

grandmother
1. 名词. 奶奶、祖母、姥姥、外祖母
Components
祖
1. ancestor; forebear; forefather
2. grandparent; grandfather or grandmother
3. founder
4. (literary) to follow the example of; to pattern after
5. . Zu
^祖 ^沖-之 - Zu Chongzhi (Chinese astronomer and mathematician during the Liu Song and Southern Qi dynasties)
母
1. mother
父母 - father and mother
母子 - mother and son
Antonyms: 父
2. (Used to address a female elder member of a family.)
姑母 - father's sister
祖母 - grandmother
Antonyms: 父
3. (of animal) female
母牛 - female cow
Synonyms: 雌
Antonyms: 公
4. pertaining to origin
母校 - alma mater

外祖母
1. n. maternal grandmother

外婆
1. n. maternal grandmother (the mother of one's mother)

姥姥
1. n. (colloquial) maternal grandmother; grandma
2. interj. (colloquial) bloody hell!; bastards!; like hell!; never!

奶奶
1. n. (chiefly Mandarin Jin Min Bei informal affectionate) paternal grandmother; gramma; granny
一個愛嘮叨dao 的奶奶nai - a nagging grandmother
今天是你奶奶nai 的生辰，所以你非來不可。 - It's your grandma's birthday today, so you've got to attend.
2. n. (Cantonese, dialectal Hakka) mother-in-law (husband's mother)
3. n. (Cantonese, Min Nan) lady; madam
奶naai5-4奶naai5-2，食得飯喇！ - Dinner is ready, ma'am!
4. n. (dialectal) breast
5. n. (Sichuanese, childish in Cantonese, Wu) milk; breast milk
6. n. (Min Bei) paternal aunt (wife of father's elder brother)

祖母
1. n. (formal) paternal grandmother
放假jià 了，我們一家大小往鄉下去探望祖母。 - We had a holiday, so our whole family went to the countryside to visit grandmother.

Dictionary entries from Wiktionary