The term fiber positioner can refer to a mechanical instrument that positions optical fibers (long strands of glass which can carry light) for use in a multi-object spectrograph, placing it so that light from the image of a particular object on the focal plane is carried to a spot along the line in a spectrograph where the light is split by wavelength. the spectrograph's slit or its line corresponding to such a slit. Thus, objects spread out over the image at the focal plane are lined up in a row so that spectrograph can spread their spectrums, resulting in a bunch of spectrums laid out in parallel, which a CCD or similar sensor can record.
For such multi-object spectrographs, this task has been carried out by hand, which is time-consuming, requires very precise work, and spectrographs now exist that can observe hundreds of objects simultaneously, demanding a lot of such work for each such observation. Mechanical positioners are a natural development, since current technology allows relatively quick and accurate placement.
Naturally the term fiber positioner can also refer to a person tasked with the positioning (like the word computer, which originally referred to someone who computes), and outside astronomy, the term can refer to devices that line up optical fibers end-to-end, such as for communication. I believe the APOGEE spectrograph used by the Sloan Digital Sky Survey, originally depended upon staff to position fibers, with cartridges holding positioned fibers that could be set up in advance and switched efficiently, so staff could prepare a number of them daily for the night's viewing. The spectrograph has been or will be automated. Some robotic fiber positioners: