The obelisk in the garden of Villa Medici is a 19th century copy of the 'Mediceo' obelisk, which now stands in the Boboli Gardens in Florence.
Dating back to the reign of the pharaoh Ramesses II (r. 1279-1213 BCE), the original obelisk was first erected in Heliopolis, one of the oldest cities of ancient Egypt. During the reign of the emperor Domitian (r. 81-96), it was transported to Rome, where it was placed in the Iseum (Temple of Isis).
At some point in the middle-ages, the obelisk toppled (or was toppled) and disappeared from view. It was unearthed in the 16th century and bought by Cardinal Ferdinand I de' Medici, who had it erected in the gardens of the Villa Medici. In 1788, on the orders of the Grand Duke Peter Leopold of Lorraine, it was taken to Florence and placed in the Giardini di Boboli.
In the 19th century the 'Mediceo' obelisk was replaced by the copy we see today.