Well, as a simple case, suppose you've got a mass positioned at location [tex]\vec{x}_{0}=(x_{0},y_{0},z_{0})[/tex] with mass [itex]m_{0}[/itex] Then, for any spatial point [tex](x,y,z)=\vec{x}\neq\vec{x}_{0}[/tex] that mass generates at that point a force per unit mass: [tex]\vec{f}(x,y,z)=-\frac{Gm_{0}}{||\vec{x}-\vec{x}_{0}||^{3}}(\vec{x}-\vec{x}_{0})[/tex] The force [itex]\vec{F}[/itex] acting upon an object of mass M situated at (x,y,z) is then found by multiplying f with M.
Another form you sometimes see assumes that the mass is at the origin, and uses spherical coordinates: [tex]\vec F (r, \theta, \phi) = - \frac{G m_0}{r^2} \hat r[/tex] where [itex]\hat r[/itex] is the unit vector in the outward radial direction at that particular point.