Is insulated copper wire required?
I am making a motor and was wondering if it is necessary to use insulated copper wire instead of any regular copper wire.


This depends on what you mean by "insulated".  

In a motor or any device that uses multiple turns to form an inductor - a device that increases the magnetic field by using a single isolated multi-turn conductor, then the current MUST flow into the wire at the input, travel through the length of the wire without any alternative paths available and exit at the other end. 


What you term "normal copper wire" is probably wire insulated with a thin conformal coating usually with a thickness in the 1-10 thorsandths of an inch. This may be polyurethane or some other 'plastic' and its thickness is normally small compared with the wire diameter. 





What you mean by insulated wire is probably wire with an outer coating of thick insulation - often PVC - with the insulated coating often coloured and able to be stripped from the wire, if desired, leaving bare uninsulated copper. The insulating coating may be 10% or more of the total wire diameter. 





Also available, and less common, is bare copper wire with NO insulating coat. It may not be visually obvious that such wire does NOT have a thin insulating coat -  testing with an Ohm meter or manufacturer's specifications may be needed to be sure. 


Both wire in 1. and 2. are potentially suited for constructing motors. The aim is to use a wire with as mechanically thin as coating as possible BUT having a voltage breakdown rating adequate for the task and adequate resistance to mechanical damage which could occur during manufacture and use. 

Wire of type 2. with thick insulation is usually used for distribution and interconnection purposes, where resistance to mechanical stress is of great importance.

Wire of type 1. is normally used in motors where voltage resistance is paramount and mechanical issues are controlled when the wire is handled and wound and in the final product.