An Inhabited Cross is any cross design that includes a picture or sculpture of some person or persons other than, or in addition to, Jesus. It might alternatively be called a Saint's Cross, Pope's Cross, or whatever identifies the inhabitant.
Of course, Christ is symbolised by the cross whether or not His image is shown (as in a crucifix) but sometimes people want to additionally revere or honour others.
These 'others' may include an image depicting an angel, a figure from the Bible, a particular saint or perhaps a family member. And the person being revered is not always deceased. The pope, for example, is often depicted with a cross. (Note: One should be careful not to say the pope or a saint is depicted on a cross, since that implies the person was crucified.)
The Inhabited Cross shown above is the same basic design as the 'Becket' Cross, with a titulus bar at the top. This particular version is the San Damiano Cross, which hung in the ruins of the church rebuilt by St. Francis of Assisi and is still to be found there. This was a popular form in the Middle Ages, and there are several different versions of the Inhabited Cross showing angels, saints and patrons of various institutions. On the cross shown above, the artist depicts people adoring Jesus in His act of redemption.
The Inhabited Cross is a development within the Byzantine artistic style imported into the north of Italy after the capital of the empire moved from Rome to Constantinople (Byzantium). It is believed the emperor's representative, the Exarch, was headquartered in Ravenna during the 5th century. The Byzantine style influenced Northern Italian religious art for centuries. The variation to the right is by Cimabue, painted for the church of Santa Croce (Holy Cross) in Florence in 1265.