Gravestone

An urn or gravestone appears on the lot when a Sim dies. An urn can be placed on the floor or on tables and counters. A gravestone can also be placed on the ground outside. In The Sims, The Sims 2, The Sims 3, and The Sims 4 whether a grave appears as an urn or a gravestone will depend on whether it is inside or outside; an urn will become a gravestone if it is moved outside, and a gravestone will become an urn if it is moved inside. Sims can "Mourn" over graves or urns. Beginning with The Sims 2: Nightlife, they can be moved to a community lot.

The Sims
In The Sims, different gravestones are available, but players have no control over which one appears after a Sim dies. Pets and famous Sims have their own urns and gravestones.

Graves cannot be placed on community lots. "The Gothic Quarter" in Old Town, and "Vernon's Vault" and "A Spooktacular Spot" in Magic Town all contain gravestones, but they are simply decorative and are not linked to any Sim's ghost.

The Sims 2
If a Sim had platinum aspiration when they died of old age, a gold gravestone/urn will appear with the dead Sim's aspiration symbol on it. Otherwise, the Sim will get the average, grey gravestone/urn.

If Nightlife or later is installed, players have the options to click on an urn or gravestone and select "Move grave(s)". This moves the grave to a community lot of the player's choice, where it will appear near the telephone. When this is done, the shade of the Sim will appear and wave goodbye, then the grave will disappear, and the player will be notified that the Sim has died on another lot and will return in spirit to the place of their death. It appears that this may refresh the tendency of family members to cry over the death, even if that had largely passed. Another way to move a grave is to place the urn or gravestone in a Sim's inventory, and send that Sim to a community lot. If player clicks on the ground and selects "Place NAME's Grave Here", the Sim will go to that spot, and the grave will appear there. Regardless of which method is used, the player will be able to select that lot from the neighborhood view, and use the hand tool in Buy or Build mode to place the graves where they wish. It is a good idea for players who plan to move graves to make a cemetery, but there is a pre-made cemetery, "Gothier Green Lawns", found Downtown next to the House of Fallen Trees.

In The Sims 2: Pets, pets get gravestones/urns that are marked with a pawprint, and the owner(s) of the pet will get a special memory. A pet's urn is grey, and a pet's gravestone is brownish.

With The Sims 2: Seasons installed, a grave can be struck by lightning during a storm. This will cause the ghost to appear (even during the day). Additionally, in Seasons or later, Sims cannot give graves as gifts.

The Sims 3
In The Sims 3, there are at least 3 different types of gravestones; a flat plaque-like one, a grey one that's similar to the ones in The Sims 2, and a tall obelisk-type one. A Sim can obtain these different gravestones by:

Small Plaque
 * Dying before earning 75,000 lifetime happiness points.

Medium Gravestone w/ Plaque
 * Dying after earning 75,000 - 149,999 lifetime happiness points.

Large Gravestone w/ Plaque engraved
 * Dying after earning +150,000 lifetime happiness points.

Note: The gravestone a Sim receives following their death is based on the total lifetime happiness points that Sim has received overall, not the number of points that are available when the Sim dies. This means that points spent on lifetime rewards do not affect this number. To see the total number of lifetime happiness points a Sim has earned, hover the cursor over the chest icon in the Lifetime Rewards panel.

A gravestone will have a plaque which shows the cause of the Sim's death:

Pets, Sims that died in the future and Plumbots  have their own urns and gravestones.

The Sims 4
In The Sims 4, ordinary Sims only have one kind of tombstone and urn. Aliens and vampires have a special kind of urn. As auto age is switched on (by default) for all played households, all played Sims will eventually succumb to death by old age if nothing else. However, only Sims who die on-screen will get a tombstone or urn (the Sim does not need to die in their own lot or be a member of the active household). Sims who die of old age off-screen do not. To ensure that all played Sims get a tombstone or urn, you will need to edit auto age to only be applied to active households.

Like in previous games, Sims can be made to mourn dead Sims (although Sims who have a significantly negative relationship with a dead Sim will be given the option to laugh at that Sims grave instead). Mourning a tombstone or urn will produce a sad moodlet that lasts for four hours.

Each tombstone has an option to 'Release Spirit to Netherworld'. Once this option has been selected, the dead Sim's remains, ghost, profile, etc. will be removed permanently from the game.

Sims who have died will by default have their spirit gradually lose connection to the physical world, and his or her spirit will automatically be released after a certain amount of time has passed. This can be halted by selecting the option to 'Strengthen Connection to Physical World' on the tombstone or urn. If this is not done, a notification will appear after a few days stating that the Sim's spirit is fading away, and suggests to the player to attempt to "Strengthen His/Her Connection to the Physical World."

Trivia

 * In The Sims 2, moving a gravestone to a community lot may change the type of death of Sims (from any kind of death) to starvation. This is also true vice versa.
 * In The Sims 3, when a Sim dies, the player has the option of whether to keep the grave on the Sim's home lot or to put it into a cemetery.
 * In The Sims 3, having a playable ghost with the same name appearing on another gravestone may cause the "Return to Netherworld" option to appear if the player clicks on it, even if the gravestone doesn't belong to the ghost.
 * The player can get gravestones and urns with buydebug cheat on known as "Remains", however, they will not contain ghosts.
 * In The Sims 3: Showtime, when a Sim dies as human statue, the gravestone has a plaque implying the Sim died by fire prior to Patch 38, where it is changed to two crossed shovels. This is also the default for remains (graves without ghosts); however, it can be changed by using cheats.
 * In The Sims 3: Island Paradise, when a Sim dies by running out of air, the gravestone has a plaque implying the Sim died by drowning.
 * In The Sims 3: Ambitions, when a SimBot dies in any way, the plaque on the gravestone will be a SimBot's hand holding up an hourglass.
 * In The Sims 3: Into the Future, Sims who die in Oasis Landing will get a futuristic holographic gravestone. However, these gravestones can be accessed by purchasing them from the buydebug.
 * Also, ghosts - who were created in Create-A-Sim - will get a holographic gravestone, if their deaths were set to be Jetpack crash or Time anomaly.
 * The holographic plaque of death by shark attack and dehydration from Island Paradise erroneously has the plaque of death by old age.
 * In The Sims 3: Into the Future, plumbots have their own gravestone and urn. The gravestone has a holographic plaque of spinning gears.
 * In The Sims 4: Get to Work, aliens have a special gravestone and urn.
 * In The Sims 4: Vampires, vampires also have a special gravestone and urn.
 * In The Sims 4: Realm of Magic, spellcasters have a special gravestone and urn.