What characteristic of a planet determines the acceleration due to gravity on it?

The mass of a planet determines the acceleration due to gravity on it. This is according to Newton's Law of Gravitation, which basically states that the more mass a body has, the greater the force of attraction it exerts on other bodies with mass near it.

The gravitational force is:

F = GMm/r², where G is a constant, r is the distance between large mass M and small mass m.

Considering the fact that acceleration is force per unit mass, if we divide gravitational force by the small mass (to get force per unit mass), we see the dependence mathematically:

a = GM/r²