Note: in the text below, we give nikkud (vowel markings) as reading help in some places. On the actual tombstones, however, most often the text does not have any nikkud at all.
A tombstone typically starts with the abbreviation פ״נ, that means פֹּה נִקְבַּר po nikbar or פֹּה נִטְמַן po nitman "here was buried". If the grave belongs to a female, it is read in feminine: נִקְבְּרָה nikbera or נִטְמְנָה nitmena.
Next goes the name of the person. If that is a man, it can be preceded with ר׳ "reb" (Mr.). Note that this by itself does not indicate a rabbinical status. If the person was a rabbi, the name will be preceded by הָרָב ha-rav "the rabbi".
Then goes the name of the person - usually, their Jewish name and not the name that was used in the secular context. It can often be a compound name, usually of two components (for example, יהודה־לייב Yehuda-Leib or שניאור־זלמן Shneur-Zalman). Sometimes it can be accompanied by a last name – and sometimes, the last name can be spelled according to the Yiddish orthography which is different from the modern Hebrew one.
The name is accompanied by a patronymic, a reference to the person's father's (or, less commonly, mother's) Jewish name. The patronymic is preceded by בֶּן־ ben- "son of" or בַּת־ bat- "daughter of". בן ר׳ "son of Mr." can be shortened to ב״ר.
Often, the name will be followed by one or several descriptive words, for example: איש ישר ish yashar, יָשָׁר yashar meaning "straight, honest", תָּם tam "honest, simple", צָנוּעַ tzanua "modest, humble", נָדִיר nadir "rare" - or something more flowery: ישר עינן ונעים הליכות yeshar einav u-neim halichot, literally "straight of his eyes and pleasant of manners".
Often, נפ׳, an abbreviation for נִפְטַר niftar, "deceased", precedes the date of death. The dates of death (and birth, if present) are given according to the Hebrew calendar. You can use any tool available online - for example, this one to convert a Hebrew date to a more familiar Gregorian date.
Note that the years denoted by Hebrew letters are pronounced as they are spelled. For example, י״ז בניסן תשע״ט (17 Nisan 5779, corresponding to 22 April 2019) can be pronounced shiv'a-asar be-nisan tav-shin-ayin-tet.
Headstone inscriptions are most often concluded with the abbreviation תנצב״ה (sometimes read as tantzava), which means תהי נפשו צרורה בצרור החיים tehi nafsho tzrura bi-tzror ha-chayim "May his soul be bound in the bundle of life", a paraphrase of 1 Samuel 25:29. If the person is female, נַפְשׁוֹ nafsho "his soul" is replaced by נַפְשָׁהּ nafsha(h) "her soul", but the abbreviation remains the same.
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