Maxwell's first equation dealt only with electric fields, saying nothing about magnetism. That changes with the introduction of Maxwell's second equation. A review of the mathematical development of the divergence, covered in detail in Chapter 2, may be helpful.
A magnet is any object that produces a magnetic field. That's a rather circular definition (and saying such is a bit of a pun, when you understand Maxwell's equations!), but it is a functional one; a magnet is most simply defined by how it functions.
As mentioned in previous chapters, technically speaking, all matter is affected by magnets. It's just that some objects are affected more than others, and we tend to define "magnetism" in terms of the more obvious behavior. An object is magnetic if it attracts certain metals such as iron, nickel, or cobalt, and if it attracts and repels (depending on its orientation) other magnets. The earliest magnets occurred naturally and were called lodestones, a name that apparently comes from the Middle English "leading stone," suggesting an early recognition of the rock's ability to point in a certain direction when suspended freely. Lodestone, by the way, is simply a magnetic form of magnetite, an ore whose name comes from the Magnesia region of Greece, which is itself a part of Thessaly in central eastern Greece bordering the Aegean Sea.