I'm far from an expert contributor here, or of the history of the Kingdom(s) of Sicily specifically but I can give you the gist of it and point you to further reading of you wanted something more specific.

Short answer: Both Sicily and Southern Italy had been conquered and unified by the Normans, who created the Kingdom of Sicily, ruled from Naples. Then Sicily revolted and went to the aragoneese, whom started to call it Sicily, understandably. The Kingdom of Sicily kept its oficial name, althou it was known as the kingdom of Naples. Both entities had their own history for centuries untill this guy King gave both kingdoms to his son, who unified them, giving himself the name "king of two sicilies".

Long: The "other Sicily" that you ask about was, well, Naples, kind of. The Kingdom of Naples was formed after the Norman conquests of Italy, and encompassed southern Italy, Sicily and some more islands, with the court at Naples. The sicilians rebelled in 1282, after such the island was ascribed to the crown of Aragon and governed from the Iberian peninsula. A couple centuries later, Alfonso V of Aragon conquered the Kingdom of Naples, still officially named "kingdom of Sicily", thus stablishing the union of the "two sicilies" under one crown, ironically, the same situation that created the one, unified Sicilian kingdom in the first place.

After a bunch of complicated centuries that saw both entities separating and briefly unifying again (while the island remained firmly in aragoneese / spanish hands, Naples changed hands a number of times, so if it was under Spanish hands the Spanish crown could adopt the title king of both sicilies, but that meant both kingdoms, not an unified one named like that, as, as I said, they were different political entities throughout late medieval and modern times) in the XVIII Century Charles III of Spain, gave both kingdoms to one of his son Ferdinand (III of Sicily, IV of Naples) under the title Ferdinand I of the Two Sicilies. After some napoleonic messing around, the Congress of Vienna gave the Kingdom back to the bourbons. It took Garibaldi to end with the unified, bourbon created and ruled Regno delle Due Sicilie; it became, as it still is, one of the more significant part of that country called Italy.