It depends on what interactions are influencing the motion of the object. In general, if you know all the forces involved, then you just solve Newton's law F = ma as a differential equation for the position.
For instance, if you have an object moving due to a uniform gravitational force F = mg, then from F = ma,
a = d2x/dt2 = g
v = dx/dt = gt + (a constant)
If we apply an initial condition that v(t=0) = v0, then v = v0 + gt.
x = v0t + 1/2 gt2 + (a constant)
Applying x(t=0) = x0, we have
x = x0 + v0t + 1/2 gt2
The trajectory is a parabola.
force = mass * acceleration.
That's all there is to it.
Let's not forget Newtons other laws. An object will continue the state it is in untill it is acted on by a force. That is if the body is moving in a straight line at a certain velocity, it will continue on the straight line at that velocity until it comes into contact with another force. A planet would travel in a straight line if it weren't for the gravitatal force of the sun continuously bending the planets path toward the sun.
Another law is that for every action there is and equal and opposite reaction. That is if press your finger on a scale until the scale reads 1 pound, you feel 1 pound of force on your finger.
These laws make mechanical motion of a body predictable if you know the forces acting on the body.