My answer will assume the reader knows some calculus, but I will try to explain the concepts in an intuitive manner as well.
Most of the question is about what we call "kinematics", that is the study of the motion of objects under the assumption the velocity/acceleration is known. Kinematics of point masses (in one dimension) boils down to these two equations:
$$ v = \dot x $$
$$ a = \dot v. $$
One can also write:
$$ a = \ddot x. $$
The dot above a variable refers to the time derivative. The time derivative is the rate of change of the quantity. In other words, in a small step of time the change in velocity is given by:
$$ \Delta v = a \Delta t $$
and the change in position is given by:
$$ \Delta x = v \Delta t. $$
So if there is a change in position, there must be a non-zero velocity. (To say, velocity causes the change in position is a bit dangerous from a philosophical point of view, rather the velocity describes that there is a change of position.)
As an example, the following situations are possible:
- The acceleration is non-zero, but the velocity is zero.
- The velocity is non-zero but the acceleration is zero (this is what we call uniform motion).
How to get the position, given the velocity in dependence of time? That is done by the integral (in a sense the inverse of the differentiation):
$$ x(t) = x(0) + \int_0^t dt'\, v(t'). $$
The same equation holds for velocity and acceleration:
$$ v(t) = v(0) + \int_0^t dt' \,a(t'). $$
Now, we will go on to the part about forces: Forces are connected to acceleration by the well known formula:
$$ F = ma. $$
So substituting this in the equations above we get:
$$ F = m\dot v = m \ddot x. $$
So force is the "cause" for acceleration, accelration is the rate of change of the velocity and if a particle has a velocity it will move.
Note, that is usually considerably harder to solve the equation $m\ddot x = F$ than to calculate the integrals for the position and velocity above. The reason is, that $F$ often depends of the position. Such an equation (where a quantity and derivatives of a quantity are related) is called differential equation.
In conclusion, acceleration is directly related to the forces acting on an object, if the forces stop to act, there is no more acceleration, but the object will continue to have its velocity (as per Newton's law of inertia) and thus continue to move. Also, applying a finite force for an infinitesimal amount of time will not change the velocity (as $a \Delta$ will be zero).