For specifying a magnet wire the following parameters must be identified:
are important for electrical properties such as resistance etc. and mechanical properties such as tensile strength, elongation etc.
Magnet Wire Types
are important for the application of the coil. For example thermal resistance (cut through temperature) or temperature durability or processing characteristics (solderability) are important criteria.
See "Magnet Wire Types".
Technical Data by Size
Many parameters vary by diameter. Most important for a specification are bare wire diameter, resistance and outer diameter. These values are necessary for coil design.