When Sims reach the end of their lives, they will die of old age.
In The Sims 2, if a Sim dies with their aspiration bar in gold or platinum, the Grim Reaper and two Hula Zombies will appear and relaxing music will play. The Reaper will flip through his notebook for a few seconds, then the Sim will pick up their suitcase and walk off with him, while fading away. Otherwise, the Grim Reaper will show the Sim their hourglass of life, and the Sim will try to pull it away from the Grim Reaper, and will eventually disappear. Old age and drowning are the only base-game death scenarios in which a living Sim will not be able to plead with the Grim Reaper to bring the Sim back from death.
In The Sims 3 death from old age happens much less dramatically, with the Grim Reaper taking Sims away. Sims with certain traits, such as Unlucky or Clumsy, can only die of old age, as the Grim Reaper finds their misfortunes too hilarious to take them any earlier, and will simply revive them. If there are other Sims not in the same household as the elder (and story progression is on) who are friends or above with the elder, they will receive a phone call when they are close to death, saying that the elder may not be around for long, and that now might be a good time to visit them. When Sims die of old age, they transform into ghosts and look at their wrists.
In The Sims 2, after a Sim who died with their aspiration bar in platinum disappears, a white grave with gold borders will appear. The symbol of the Sim's aspiration, colored gold, will also be on the grave. Otherwise, it will be a normal, grey grave. However, a player should be warned, that in The Sims: Castaway Stories there seems to be a glitch with these types of golden graves: the tombstones are stuck, invisible objects (while the golden urns are okay).
In The Sims 3, the size of the tombstone is determined by whether the Sim completed their Lifetime Wish and how many Lifetime Happiness Points they had at the time of death. This applies to any form of death in The Sims 3.