Mummies were introduced with the World Adventures expansion pack.  They live in Al Simhara (the Egyptian world), inhabiting underground tombs.  

Finding A Mummy

To find a mummy, look for a sarcophagus.  They sleep there - all the time, apparently!  (Mummies must be even lazier than I am.)  You can awaken the mummy by right-clicking on the sarcophagus and instructing your Sim to look inside.  Mummies may also spontaneously awaken if you start stealing treasure from their tomb.

Fighting The Mummy

Once the mummy awakens, it will pick a fight with your Sim.  If your Sim loses this fight, they will receive the mummy's curse.  

There are two ways to win your fight with the mummy: if your Sim has a high enough martial arts level, they will win the fight outright.  Or, if your Sim happens to have the Mummy Snack in their inventory, the mummy will snatch the snack and return peacefully to their sarcophagus.  (In other words, the Mummy Snack serves the same purpose with mummies as the Death Flower does with the Grim Reaper.)

Removing The Mummy's Curse From Your Sim

In the Sims 3, if you get cursed by a mummy, you have two weeks to live.  If you don't remove the curse in that time, you will die.  

To remove the curse, you can either pray at the statue inside the sphinx, or rebuild both of the King's Sarcophaguses (both Blessed and Cursed).  Sleep in the Cursed sarcophagus to turn into a mummy, then sleep in the Blessed sarcophagus to be turned back to human - and without the curse.  

Turning Your Sim Into A Mummy

Your Sim will turn into a mummy if they spend enough time sleeping in the King's Cursed Sarcophagus.  They have to sleep for at least two sessions, of several hours each.  When your Sim emerges from the sarcophagus the second time, they will have been turned into a mummy.

Life as a Sims 3 Mummy

There are advantages and disadvantages to being a mummy.  Mummies are very strong in comparison with regular Sims, they live a little longer, and they do not have the Energy or Bladder motive.  And if you are a mummy, you can explore the Tombs with impunity, because other mummies will not attack you.

On the down side, mummies are really flammable!  It's all those old rags and bandages, you see.  They go up like a candle if you get too close to flames.  In fact, while it can take a long time for a Sim to die in a fire, a mummy will die instantly.  Thus, they get the "Scared" moodlet whenever they are around open flame (like a lit fireplace).  

Mummies also move with a very slow, shambling gait, which can be frustrating for the player.  And they are unable to conceive children.  Female mummies cannot get pregnant, and male mummies cannot impregnate a female.  However, mummies can fall in love, get jobs, get married, and do all the other regular bits of Sim life.