Force and torque of a rod

It will rotate as well as translate because the force is not applied to the center of mass and there are no other forces (except gravity, which is downward) acting on the rod. If the force was applied to the center of mass of the rod (presumed to be at the center of the rod if its mass is uniformly distributed along it length), then the rod will only translate and not rotate.

As you explained that for the case stated in the question the rod will translate and rotate, it would be a little bit more helpful for me if you could explain why does it rotate.

See the diagram below.

For the purpose of determining rotation, the center of mass of the rod can be considered mid point of its length (assuming its mass is uniformly distributed along its length.

You can see there is a net torque of $tau=FL/2$ about the center of mass which will initiate clockwise rotation rotation of the rod. The force $F$, which is unopposed, will also result in translation of the COM. If the line of action of the force was through the COM, there would be no net torque on the rod and its motion would only be translation.

Hope this helps.