Sodium is produced by electrolysis of molten sodium chloride. what are the products at the anode and cathode, respectively?

Answer is: chlorine and sodium.

Molten sodium chloride is separeted::
2NaCl(l) → 2Na(l) + Cl₂(g), 
but first ionic bonds in this salt are separeted because of heat: 
NaCl(l) → Na⁺(l) + Cl⁻(l).

Reaction of reduction at cathode(-): Na⁺(l) + e⁻ → Na⁰(l) /×2.

2Na⁺(l) + 2e⁻ → 2Na(l)

Reaction of oxidation at anode(+): 2Cl⁻(l) → Cl₂(g) + 2e⁻.

The anode is positive and the cathode is negative.