I'm not sure what do you mean by simple electromagnet. The answer can vary from one person to other. The DIY electromagnets are the simplest ones which can be made by wrapping an electrical wire around an iron core such as a nail, and connecting both ends of the wire to a battery. This should be enough to design a working electromagnet.
If you go for more sophisticated applications like motors, generators, etc. the shape of the core will change. The core usually consists of loops.