Gershayim most commonly refers to the punctuation mark〈״〉. It is always written before the last letter of the non-inflected form of a word or numeral. It's used in the following ways:
To indicate an acronym. For example: דּוּ״חַ (singular), דּוּ״חוֹת (plural), "report" represents דין וחשבון; and מ״כ (masculine), מַ״כִּית (feminine), "squad commander" represents מפקד כיתה.
To indicate a multi-digit Hebrew numeral. For example: ח״י represents 18. Single-digit numerals are indicated with a following geresh.
To indicate the names of Hebrew letters, differentiating them from any homographs. Compare הוּא שִׂרְטֵט עַיִן "he sketched an eye" with הוּא שִׂרְטֵט עַיִ״ן "he sketched an ayin".
To indicate Hebrew word roots. For example: the root of תַּשְׁבֵּצִים /taʃbeˈtsim/ "crossword puzzles" is שב״צ (š—b—ṣ); the root of לְהַטּוֹת /lehaˈtot/ "to tilt, to conjugate" is נט״ה (n—ṭ—h); and the root of הִסְתַּנְכְּרְנוּת /histankreˈnut/ "becoming synchronized" is סנכר״נ (s–n–k–r–n).
In older texts, to indicate the transliteration of a foreign word. This use corresponds to English's use of italics. For example: in printed works of Rashi, the town of Rashi's birth, Troyes, is spelled טרוי״ש.