Ghosts have been a part of the game ever since the first Sims was released.  In The Sims, ghosts would scare your Sims until you deleted their tombstones.  (Or had your Sims sleep on the second floor, while the tombstone or urn was on the first floor.  Ghosts didn't climb stairs.)  Sims 2 introduced ghosts with increased functionality, and different colors depending on the mode of death.

Sims 3 has taken another step forward and endowed ghosts with a full personality.  If your Sim travels to the graveyard at night, he or she can interact with the ghosts, holding conversations just as they would with another (living) Sim.  Sims can even start a romance with a ghost, get married, make Woo Hoo, and have a ghost baby!

According to the Sims Wiki, a baby your Sim has with a ghost will be either a normal baby or a ghost baby.  Ghost babies are transparent, carry their own headstone in their inventory, age like a normal Sim baby, and continue to exist during the day.

You may have seen Ambrosia listed as a buyable recipe at the bookstore.  Ambrosia requires level 10 Cooking skill to create, level 7 Fishing skill to catch the Death Fish, and level 7 Gardening skill to grow the Life Fruit.  However, if you manage to whip up a batch of Ambrosia, feeding it to a ghost will restore them back to life.

(Feeding Ambrosia to a living Sim will max out their Hunger bar, lift their mood by 75 for a week, and restore youth by bringing their age back to the beginning of their life stage bar.  Not too bad, but I still miss the Cow Plant, it was so easy!)

Ghosts can also become playable characters!  After a member of your household dies, within a few days one of your Sims will get an opportunity to take the urn to the Science Lab for a resurrection experiment.  Sometimes the experiment is successful, and the Sim is restored to life and the household.  Other times the experiment is only half successful, and the Sim is returned to the household as a ghost.

When this happens, you can control the ghost Sim just like you would any other Sim.  In addition to the regular actions, they will be able to "frighten" people.  Ghosts are second class citizens in Sims 3, and are unable to become partners in or buy out businesses.  However, they can lead happy and fulfilled lives.  (No word yet on whether a playable ghost eventually dies, or if they "live" forever on their lot.)

Regarding the resurrection: you can't just take an urn to the Science Lab, you have to wait for the Opportunity to pop up.  However, the Sim who accepts the resurrection opportunity and takes the urn to the Science Lab will gain the ability to resurrect ghosts at the Lab at any time in the future.  I guess once you've been invited to bring over an urn, they let you drop by with a fresh urn any time!