What Kinds Of Ions Do A Sodium Atom And A Chlorine Atom Become When A Valence Electron Is Transferred From One To The Other?

Sodium is a metal and an alkali element. This means that it loses its electrons during a chemical reaction.
Chlorine on the other hand is a halogen and from Group 7. These compounds gain an electron during a chemical reaction to become stable and  complete their  octet for the last electron shell.

Let's look at their reactions:

Na ---> Na⁺ + e^-

Chlorine takes up the electron lost by sodium:

Cl + e^- ---> Cl^-

So, when they combine, the resulting compound, sodium chloride, has a lattice of positive ions of sodium called cations and negatively charged chloride ions called anions.

2Na + Cl₂ ---> 2 Na⁺Cl^-