A Burial Cross, Grave Cross or Rest In Peace (R.I.P.) Cross marks the place where somebody(!) is buried.
For sure, when we die, we don't smell very nice. (Many of us don't smell so nice when alive, either1.) This is nature's way of protecting the living, warning us away from possible infectious disease by making decomposing corpses unpleasant to have around. Since Adam's time, we have dug holes, deposited bodies, and covered them up.
Burial is also a mark of respect for the deceased. The grave is the last resting place of the physical person and its precise location is important to family and friends. Consequently, a stone marker is used in most cultures that bury their dead.
For Christian cultures, the tradition has been to use a stone cross marker. The cross reminds the bereaved that the soul does indeed, live on into eternity.
The modern cross above (left) was cut from marble; a relatively easy process with today's modern machinery. Each arm has a Fleur de Lis and the inscription is IHS. Earlier grave markers, reflecting the limitation of machinery and perhaps scarcity of masons, were much simpler and showed just the deceased's initials (above right).
The cross is so ubiquitous in graveyards that it has become a symbol for death, as shown in the advertising campaign to reduce driving speed.
In fact, the Christian cross is really a symbol of life (but it's a neat speed-kills advertisement nonetheless).
After death, Christians (and many of those with other faiths) believe that the Messiah will return and take his followers to Paradise. The Bible details this quite clearly, even to the point of assuring us that Christ will return from the east, which is why traditionally people are buried with their feet pointing eastward so they can be raised2 facing Christ to welcome him on his return.