There are indeed no particles with “magnetic charge” in the same sense that there are particles with electric charge. Technically speaking, this means that magnetic fields are divergence-free–any (imaginary) closed surface has just as many magnetic field lines going into it as going out of it. Contrast this to the case of electric fields, where the difference between incoming and outgoing field lines is directly proportional to the charge enclosed by the surface (Gauss’ Law).
Electric and magnetic fields are closely related, however (hence the term electromagnetism used to describe the phenomena associated with them). In particular, electric fields that change with time (and moving electric charges, which amount to the same thing) generate magnetic fields, and vice-versa (see Maxwell’s Equations in any intorductory physics text for an idea of the relationships between electric and magnetic fields and charge/current. Electromagnetic fields are not associated with the vibration of any medium–the concept that light waves were carried by the “ether” was disproved by the Michelson-Morley experiment more than 100 years ago.
Currents and time-varying electric fields are not the only way to generate magnetic fields, however. Particles such as electrons, protons, and neutrons have intrinsic magnetic moments (i.e. they act like tiny bar magnets). Note that these are not magnetic charges, since the dipole fields still have zero divergence. The magnetism of the metals like iron, cobalt, and nickel comes from the orbital angular momentum of unpaired d-shell electrons (IIRC) which generates a dipole field like a little current loop (Did you ever make an electromagnet by winding a wire around a nail? It’s a little like that.).
As for magnons, I’m not sure, but I assume they are quantized magnetic flux excitations in a solid material. And you probably don’t want me to get into flux quantization (besides, it’s been about 10 years since I even looked at this stuff, so I’m extremely rusty). Any solid state physicists out there?