Does an aspheric lens require that the incident light be collimated in order to focus it? Aspheric lenses are able to either collimate or focus light, depending on the side of the lens the light is incident. In diagrams that show the aspheric lens focusing light, the incident light rays always look to be collimated. Does an aspheric lens require that the incident light be collimated in order to focus it? If so, then why?
 A: Yes it will focus an uncollimated beam, however, the focus will not be at the specified focal point, which is specified for collimated beams. You essentially have a two lens system where the focal point of the first $f_1$ is less than the distance between lenses, this gives a new focal length of
$$\frac{1}{f_b}=\frac{1}{f_2}-\frac{1}{\alpha},$$
where $f_b$ is the new focal length, $\alpha$ the distance from the second lens to the focal point of the first and $f_2$ the focal point of the second lens. In your scenario you don't have the first lens, the focal point of the first lens is replaced by the uncollimated beam source, and the distance from the beam source to the lens (your aspheric lens) now $\alpha$. The difference between aspheric lenses and spherical being that aspheric lenses produce a sharper focal point.
In the lab, it is quite common that we come across unlabelled lenses. We check the focal point with a flash light far away from the lens $1/\alpha\approx 0$ (collimated beam) and move the lens towards/away from the wall until we see the focus. Also note, that if $\alpha=f_2$ then $f_b=\infty$, the condition of a collimated beam.
A: Note, lens designs are complex and there is a large variety of shapes acoording to the purpose the lens has. My answer below assumes a aspheric lens to replace a usual biconvex lens for the purpose of overcoming spherical abberations:
Every lens can focus or collimate light, but in a normal lens you have spherical abberation. That means, if you have incoming parallel light, the rays farther away from the optical axis (hitting the lens towards the edges) have a slightly different focal point than rays closer to the optical axis. So your focus point is good but smeared, and that leads to smeared images if you use the lens in imaging systems, e.g. in a camera.
Aspherical lenses are specially designed to overcome this and to produce a sharp focus point, meaning all rays independent of their lateral distance to the optical axis are focussed in the same point. The result is a sharper camera image for instance. You achieve this by specially design the curvature of the lens, and thus it is important from which side the incomming parallel rays are.
Don't get that wrong, the lens works independent of its orientation, but in one orientation the abberation is reduced. So it focusses parallel light and collimates light from a single source, the orientation is important for compensating spherical abberation.
