What is the molecular polarity of NCl3 and why?

It is polar, because its charges are distributed asymmetrically and its geometric shape is asymmetrical.

Polar Molecules: 2 or more lines of symmetry when you cut through the middle of the molecule; symmetrical geometric shape and symmetrical distribution of charge! Nonpolar Molecules: Less than 2 lines of symmetry when you cut through the middle of the molecule; asymmetrical geometric shape and asymmetrical distribution of charge!