Magnet wire is simply the name given to enamel coated solid strand wire (enamel is not the only material used for this anymore).
Most wire is coated with a thick plastic type insulation and can be stranded or solid. Wire used for making electromagnets for motors, speakers, generators, etc. needs to be packed as tight as possble to get the most intense magnetic effect. Therefore the insulation is simply a layer of "paint".
Other wires are stranded to make them more flexible (not needed or wanted for an electromagnet) and have much more durable insulation so you can step on it, run over it with the vacuum cleaner, pinch it in a hinge, etc.
Any wire can be used to make an electromagnet, "Magnet wire" is optimized for it.
Hope this helps