It is an ancient Jewish custom to erect a Matzeivah / מצבה — gravestone in memory and in honor of the deceased, as we find in the Book of Genesis 35:20: “And Jacob placed a monument upon her grave: that is the monument of Rachel's grave unto this very day / ויצב יעקב מצבה על קברתה, הוא מצבת קברת רחל עד היום”. The monument is also called Tziyun / ציון — a marker (Kings II 23:17) or Nefesh / נפש — a soul, referring to the fact that the Nefesh of the deceased remains constantly hovering over the grave, which we mark to give the Nefesh a defined place to dwell.
Tombstone design. The tombstone is made from marble or granite and is usually placed at the head of the grave, but often covers the whole grave. It is better to erect a modest monument in order not to embarrass those who cannot afford an ornate headstone. The soul of the deceased would be more pleased if the mourners would donate money to charity, instead of building extravagant and expensive monuments. However, one should ignore any prior request of a deceased parent to not erect any monument at all. In this case a small tombstone should be placed on the grave. Small plaques that are set flush with the ground are not recommended, though, because it is hard to see the grave boundaries they mark and people will end up stepping on a grave.
Tombstone inscriptions. Ashkenazim engrave the Jewish name of the deceased and his or her father's name on the gravestone, but the prevalent custom among Sephardim is to write the Jewish name of the deceased and his or her mother's name instead. Some only write the Jewish name of the deceased and the family name. The date of passing that is engraved on the monument should be the Jewish calendar date, not the civil calendar date. Any additional inscriptions of praise for the deceased may follow in Hebrew or in the vernacular, but one should not write excessive praises on the monument.
The custom is to start the inscription with the Hebrew acronym פ"נ, which stands for “Here is buried / פה נקבר, פה נטמן”, and to end it with another acronym תנצבה, which stands for “May his/her soul be bound in the bundle of life / תהיה נפשו/נפשה צרורה בצרור החיים”. This phrase is based on the verse in the Book of Samuel I 25:29“May my master’s soul be bound in the bundle of life with the Lord your God / והיתה נפש אדני צרורה בצרור החיים את ה' אלהיך”.
Certain stylized Jewish symbols, such as the Menorah / המנורה, Magen David / מגן דוד, Torah scroll, Lion of Judah, etc., are frequently engraved on a tombstone. For a Cohen / כהן it is customary to engrave the hands spread in the priestly blessing. However, engraving the form of a human being and placing any pictures on the monument is not considered appropriate. It is permissible, but preferable not to erect a double monument if one of the spouses is still alive, for that shows lack of belief in Mashiach / משיח — the Redeemer, who may come any day.