Does earths gravitational force vary depending on where you are on the planet? Does the gravitational force acting on a person ever change just by their position on the planet. For instance, at greater altitudes or closer to the equator.
 A: The acceleration due to gravity changes not only on the surface of the Earth (depending on where you are) but also how high up you are (which varies by $\frac{1}{r^{2}}$, where $r$ is the distance from the center of the Earth to you). 
For more information on how it varies depending on location, maybe this will be of use?
GOCE Delivers Best Gravity Map of Earth Ever
A: Most certainly it does: the variation can be measured by a sensitive acceleration called a Gravimeter (see Wikipedia page with this name) and is the basis for gathering data important for minerals exploration. Bodies of mineral ore distort the Earth's gravity and thus can be found by measuring the variation of the local gravitation as a function of position.
Let's define for this answer "gravity" to be the net acceleration of someone sitting on the Earth (i.e. stationary relative to their Earthly surroundings) relative to their momentarily co-moving inertial frame (i.e. the frame with the same tangent at time $t=0$ to spacetime as the observer but which, unlike the observer, is free to move along spacetime geodesics). This co-moving inertial frame would be heading through the observer's feet straight towards the Earth's centre on a perfectly spherical, nonspinning Earth. 
So the total acceleration one will perceive in standing still on the Earth is thus a mixture of the "curvature" of spacetime wrought by the Earth itself and also the effect that Mike Dunlavey talks about in his comment:

If you are on the equator, you are lighter than you are at the poles, not because gravity is weaker, but because you are traveling eastward in a big circle around the earth at about 1600 kph. That also makes the earth itself bulge at the equator

This latter effect distorts the Earth's shape, and thus the "intrinsic" spacetime curvature effect as well as the effect of whirling eastwards as in the comment are both felt.
So there are both "large scale" variations owing to the Earth's bulge - from a Newtonian standpoint this is just the variation in $1/r^2$ as in Sakanojo's answer, but there are also well measurable short range effects whose measurement is used in minerals exploration. Local strengths also measurably change as you shift from hills into dales and contrawise.
