What primary factor dictates the exact path a star traces across the sky relative to an observer's latitude?

The star's declination, which is equivalent to its celestial latitude.

The precise path a star appears to follow—whether it rises and sets, remains circumpolar, or never rises—is fundamentally determined by its declination. Declination is the celestial coordinate analogous to latitude on Earth, measuring how far north or south a star is positioned relative to the Celestial Equator. Stars with high positive declination (far north) trace small circles around the North Celestial Pole, while stars with low or negative declination (far south) may appear below the horizon entirely for northern observers. This dependency means the geometric appearance of the star's nightly arc is mapped directly by its declination relative to the observer's own terrestrial latitude.