Answer:
   
   In the Torah, we read that Jacob set up a marker for Rachel (Genesis
   35:20). This led to the practice whereby Jewish graves are marked with
   the name of the deceased. Rabban Gamaliel's instructions for burial
   emphasized equality and simplicity (which is a hallmark of the Jewish
   burial customs); thus, large ornate stone markers are discouraged. His
   son, Rabbi Simeon ben Gamaliel is quoted in Beraishit Rabbah (82:11)
   as saying, "We need not erect monuments for the righteous; their
   accomplishments are their memorials." In fact, stone markers were not
   normative until the Middle Ages; Rabbi Solomon Adret (13th century,
   Spain) prescribed the use of a matzeivah (burial marker). Nowadays, tt
   has become traditional to mark a grave with a stone monument or metal
   plate on the ground. This is generally done some time during the first
   year, prior to the Yahrzeit (first year anniversary of the death), but
   traditions differ widely.
   
   The dedication of the marker is a rather late tradition of American
   Jewry (19th century). Now, it is widely done and carried over to other
   countries, including Israel. The tradition is that the dedication is
   done at the end of the Avelut (mourning) period or 11 months following
   the death. It is an act of spiritual closure ending the time of
   recitation of the Kaddish prayer for a loved one. It is traditional
   not to return to the cemetery for 30 days (Sh'loshim). Therefore, one
   would not even order a marker until after that period, assuming the
   mourner would want to compare stones and inscriptions, etc.
   
   Israelis occasionally dedicate the headstone (Matzevah) at the end of
   the Shiva (7 day) period. The reason I have been told that Israelis do
   a quick unveiling is that family might have traveled far (e.g. from
   outside Israel) for the funeral and it would be too expensive to
   return 11 months later. A problem for such a quick unveiling is that
   the stone cutters cannot prepare the stone in time for such a quick
   ceremony.
   
   In many communities outside of the United States, the unveiling is
   often done after 30 days. Some Sephardim do return to the cemetery and
   have a ceremony marking the end of shiva. Their burial customs vary
   with those of Ashkenazim.
   
   The unveiling ceremony itself is a simple graveside religious service
   marking the formal setting of a loved one's headstone at the cemetery.
   It is a brief ceremony, with a few psalms, an actual unveiling of the
   stone, and the Kaddish. The presence of a rabbi or cantor is not
   required.