Knowing that the adjective palatial is derived from the same Latin word as palace gives you a good sense of its meaning: magnificent, reminiscent of a home fit for a king.
The Palatine (Collis Palatium in Latin) is the center of Rome’s famed Seven Hills. Tradition holds that it was the location on which Romulus founded the city, and it became a prestigious site for powerful Romans—including emperors—to build large, lavish homes. Palatium made its way into every Romance language and beyond: it became palazzo in Italian, palacio in Spanish, and palast in German. Palace entered English via the Old French palais. Power, affluence, extravagance: these are the qualities that should come to mind when you encounter the word palatial.