What is the relation – if any – between the force acting on an object and the direction in which the object is moving?

Direction of the object is in the direction of the force acting on the object, but it is always not true, because the acceleration vector is not necessarily is in the direction of velocity vector.
When a car is moving with constant velocity and if you apply the break suddenly, then the car decelerate. The direction of velocity is in the forward direction but the acceleration is in just opposite to the direction of the velocity (backward direction). Therefore the direction of force will be in the backward direction in this case. From the above observation we conclude that, there is no relation between the force acting on an object and the direction in which the object is moving.