this is a coin of the Papal States. The Papal States were in 1860 a state in decline. Pius IX, who was with 32 years one of the longest reigning popes ever, had to leave Rome in 1849 after a revolution which led to the Roman Republic. after intervention of France, austria and Sicily, he was restored the same year. His restoration however didn't last long: the unification of Italy stole most of his territories. In 1859 the papal territories Emilia and Umbria voted for union with the new kingdom Italy, so the pope was left with only his heartland: Latium. After a cry for help in catholic countries, the Zouaves regiment was formed to protect the Papal State, but even that couldn't save the state. In 1870 he lost even his heartland and his capital Rome to the new state. Even when Italy granted the Vatican with a special status, Pius IX (and his successors) volutarily "imprisoned" themselves in the Vatican.
Numismatically speaking, your coin was issued under the old monetary system : 5 quattrini = 1 baiocco, 5 baiocchi = 1 grosso, 6 grossi = 1 trestone. Forced by the surrounding new state of Italy, the pope was forced to change the monetary system to a decimal system in the 18sixties: 5 centesimi = 1 soldo, 20 soldi = 1 lira.