Hello. I am a scientist here at Fermilab and your question got forwarded to me. In some sense it is a question with an easy answer, but like most science, you can keep probing the answer until you reach a question that can't be answered. I'll give it a try:
An electromagnet uses a current of electricity to form a magnetic field. Usually this magnetic field is 'funnelled' and concentrated through iron or some other magnetic material. The basic properties of electromagnetism were determined first by Ampere & Oersted in the 1820's and then by the great Faraday in the 1840's. These guys were experimentalists and they found that a changing electric field (like a moving electron in a wire) can cause a magnetic field and that a changing magnetic field (like a dynamo moved by a waterfall) can create an electric field. The two types of fields are intricately coupled. Thus, if a changing electric field causes a certain level of magnetic field, then twice that electric field causes twice the magnetic field. The mathematics of electromagnetism were set down in a very nice mathematical form by James Clerk Maxwell in the 1860's and we still use them today. However, the big question is: what is a field? Today we regard fields as related to exchanges of fundamental particles like we study at Fermilab. However, even with all my study of physics, I'm afraid I can't really explain what a field 'really' is.
Hope this helps. Feel free to ask me about anything else you are concerned about.
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