If a body is lying on a frictionless surface, and we give it an impulse, will it start rotating about its center of mass? If not, about what point will it rotate? If we want to know about what point an object will rotate in questions like these, how do we figure it out?
 A: It depends: if you give it an impulse aligned with the center of mass, then it will not rotate
However, if the impulse is not aligned (which is what always happens, because impulses are never fine enough), then it will also rotate
It will rotate about the center of mass, of course, because we always like to describe rotations around that point.
However, you can also choose another point. If a body rotates, it rotates about any point. The "rotation" will be non uniform and more complex if you choose a point different to the center of mass.
So in su the movement will be a global traslation plus a rotation about the CoM.
The concrete movement can be derived using equations: Newton's Laws of motion, moment of impulse for angular momentum and so on.
A: Consider the free body diagram:

Just one force $F$ acts on the body.
To work out whether the object will translate or not, apply $\text{N2L}$, in both $x$ and $y$ directions:
$$F_x=-F\cos\theta=ma_x$$
$$F_y=F\sin\theta=ma_y$$
So there's acceleration ($a$) in both senses.
There's also torque $\tau$ about the CoG, if $\delta$ is the perpendicular distance between the force line and the CoG, then:
$$\tau=-F\delta=I\alpha$$
where $I$ is the moment of inertia about the CoG and $\alpha$ the angular acceleration $\frac{\text{d}\omega}{\text{d}t}$. So here there would be counter-clockwise rotation.
For the more general case where there are more than one force acting, the net forces $\Sigma F_x$ and $\Sigma F_y$ and the net torque need to be determined to apply $\text{N2L}$.
