It depends on the mass of the planet and the diameter. Since planets are usually sphere shaped, it effectively means that the gravity grows with diameter (for same density) and vice versa (for same diameter, grows if density grows). Moon is about 4 times smaller (for same density, gravity would be about 4 times smaller) but Moon has lower density as well (about 60% of Earth) so it comes down to those 16%.
Rough napkin formulas below. Apologies for formatting)
F=ma
F(g) = GMm/(r^2)
ma=GMm/(r^2)
a=GM/r^2
Volume of a sphere = (4/3)pir^3
mass of a planet = density volume
M=ro(4/3)pir^3
a=(Gro(4/3)pir^3)/r^2
a=Gro(4/3)pi*r
given that G, 4/3 and pi are constants, it comes just to density multiplied with diameter.
Higher density allows the surface to be closer to the center of mass, so the gravity there will be higher. Stand on a hovering platform 6300km from the Moons center and the gravity will be just 1% of Earths, since distance beeing equal, the only deciding factor will be the mass.
That's why the gravity on the surface of neutron stars is so high, even when the mass is the same as our Sun.
The earth doesn't have uniform density at all depths - not sure about the moon. I forget exactly, but I was just reading about how if you could travel to the earth's core, because the density increases a lot, you would continue to experience around 1G something like halfway to the center.
