Magnetism is a broad umbrella term that signifies an array of physical phenomena exhibited by materials subject to a magnetic field. All materials are influenced to some degree by the presence of a magnetic field, but in many the effect is slight.
Some materials are attracted to a magnetic field. This property is known as paramagnetism. Its opposite is diamagnetism. Those comparatively rare materials that exhibit this property are repelled by a magnetic field.
Permeability is represented by the lower-case Greek letter Mu, μ. It is a measure of the degree to which any material will exhibit an auxiliary magnetic field denoted by the letter H, when in the presence of an applied magnetic field denoted by the letter B. Accordingly, the basic formula is:
B = μH
One example of a highly permeable material is a common (not hardened) nail. It is composed of soft iron and can be easily bent without snapping.
Referring to the above formula, the higher the value of μ, considering a specific material, the greater its attraction to a permanent magnet. B depends upon the values of both H and μ.
The magnetic permeability of vacuum is 0. Materials with a permeability of less than μ0 are diamagnetic, meaning that they are repelled by a magnetic field. Examples are bismuth, antinomy and superconductors. In general, diamagnetism is a weak force. It is present in all materials, but greatly overwhelmed by paramagnetism in all but a few types of matter.
Inductance is a property of any conductor. If the conductor is bent at all from a straight line, the inductance rises. It may be multiplied by winding the conductor in a continuous coil. Inductance may also be greatly increased by forming the coil around a core that has high permeability. This is generally done (and the soft iron core is laminated to prevent harmful circulating currents) for magnetics such as motors, relays and transformers. Sometimes an air core is used to intentionally limit the inductance, as in RF coils.
Good magnetic core material must have high permeability. Permeability also varies with magnetic field. Values found in tables are usually given for a zero frequency; in practice, permeability is generally a function of the frequency. When frequency is considered, the permeability can be complex, corresponding to an in-phase and out of-phase response.
At high frequencies permeability is said to have a complex permeability. At low frequencies in a linear material the magnetic field and the auxiliary magnetic field are simply proportional to each other. At high frequencies these quantities will react to each other with some lag time. The resulting fields can be written as phasors where the permeability becomes a complex number. The ratio of the imaginary to the real part of the complex permeability is called the loss tangent, which provides a measure of how much power is lost in a material versus how much is stored.