The “Famous Painting” is of course the Mona Lisa. It was painted by Leonardo Da Vinci in the early 1500s, probably started around 1503 and finished around 1517. As the Animal Crossing name suggests, it is probably the most famous painting in the world and the epitome of the Italian Renaissance. The painting is believed to be of Italian noblewoman Lisa del Giocondo. It was left by Da Vinci to his assistant Salaì, who sold it to King Francis I of France. It remained the property of French royalty until the French Revolution, after which it went on permanent display at the Louvre.
The original Mona Lisa has been located in the Louvre Museum since 1797. If you're planning a visit to Paris, France, it's a must-see, along with all the other treasures at the Louvre. Visit their official museum web site for admission details.
You can't buy the Mona Lisa IRL, of course, but Handmade Arts Limited sells an impressive oil-painted replica for about $900. If that's too rich for your blood, you can get a pretty nice canvas print by Decorarts on Amazon for under $60.