The shaft itself is called a spindle, but also, in shop-floor practice, the word often is used metonymically to refer to the entire rotary unit, including not only the shaft itself, but its bearings and anything attached to it (chuck, etc.).
Materials that spindles work with include embroidery, foam, glass, wood, etc.
Some machine tools that specialize in high-volume mass production have a group of 4, 6, or even more main spindles.
Both types, the belt-driven and the integral motor spindles, have advantages and disadvantages according to their design.
Which one is more desirable depends on the purpose of the machine and product(s) being produced.