We completed a lab and the conclusion write up asks us to describe the purity of the product. What are some things that determine the purity of the product?

Not much to go on here. Is it inorganic or organic?

Color in some cases
Analysis for constituent elements which I expect you did not do.
density if liquid (also melting point and boiling point)
melting point if solid
Reactions with product to see if you obtain the expected product.
That's about as specific as I can get for such a general question.