To begin with you need materials that transmit optically in the wavelength range you're interested in. So polymer fibres (PMMA for instance) have a different wavelength window to glass (silica) fibres. In order that the light is guided adequately you then need a higher refractive index in the core to the cladding. This is conveniently achieved by doping the core to raise it's index although the reverse can be done (but that means doping the bulk of the product - so is usually inefficient unless only a small part of the cladding is doped.
Do not forget the mechanical properties that the fibre needs. An example is found with the early composite 'fluoride' glasses ('ZBLAN' being one type') that had a very low optical attenuation but were sensitive to moisture and cracked in the real world.
This the basic and fundamental idea to choose the material for optical fiber or cladding that the material has tenancy to occupy the signal in the axial region because this reduce the loss of signal.
Both of the above answers are correct, but to put some nuance into Mr. Singh's answer, the index difference in the core and cladding determines the transverse modal structure of the fiber. If you are interested in a single mode fiber, which is often the case in fiber design for long-haul transmission, then the index difference must guarantee that higher order modes are cut off. In this case, the mode will inevitably extend into the cladding. And in this case, the losses in both core and cladding are very important as is scattering from the core-cladding interface. These considerations determine the indices, relative index and real/imaginary parts of the index of both core and cladding (both of which might comprise multiple layers).
Arvind simple asked what is the importance of material choosing in the optical fiber. You explain more , I think out of point of reference. Materials based optical fiber provided the facility to reduce the phase velocity in the material and give a unique direction to centralize the propagation of the signal in a one direction. There are less chances to scattering the signal from the core part of the structure.
David this is the optical fiber terminology not any history explanation.
There are three main factors to be considered while choosing materials for optical fiber production. Optical loss, mechanical strengths, and manufacturability of the material with existing fiber production processes. For low loss and high strength fibers, till date, the silica glass proved the best material with good manufacturability. If in some applications, the loss is not a prime issue, plastic fiber using PMMA polymer is recommended (Refer to BOOK: Fundamentals of Plastic Optical Fibers, by Yasuhiro Koike, 2015 Wiley-VCH Verlag GmbH & Co.