What is the difference between the magnetic H field and the B field?

H is the driving force in coils and is ampere turns per metre where the metre part is the length of the magnetic circuit. In a transformer it's easy to determine this length because 99% of the flux is contained in the core. A coil with an air core is difficult as you might imagine.

I think of B as a by-product of H and B is made bigger by the permeability of the core.

In electrostatics, E (electric field strength) is the equivalent of H (magnetic field strength) and it's somewhat easier to visualize. Its units are volts per metre and also gives rise to another quantity, electric flux density (D) when multiplied by the permittivity of the material in which it exists: -

\$\dfrac{B}{H} = \mu_0\mu_R\$ and

\$\dfrac{D}{E} = \epsilon_0\epsilon_R\$

Regarding ferrite data sheets, the BH curve is the important one - it tells you the permeability of the material and this directly relates to how much inductance you can get for one turn of wire.

It will also indicate how much energy could be lost when reversing the magnetic field - this of course will always happen when ac driven - not all the domains in the ferrite return to produce an average of zero magnetism when the current is removed and when reversing the current the remaining domains need to be neutralized before the core magnetism goes negative - this requires a small amount of energy on most ferrites and gives rise to the term hysteresis loss.

Other important graphs in a ferrite data sheet are the permeability versus frequency graph and permeability versus temperature.

From personal experience of having designed a few transformers, I find them tortuous in that I never seem to naturally remember anything other than the basics each time I begin a new design and this is annoying - in this answer I had to double check everything except the units of H!

Short version: Both B and H come from either magnets or current.

One (H) is straight "ampere turns", (no : Andy is correct : ampere-turns per metre) the other (B) is H times the permeability of the magnetic circuit. For air or vacuum, this is 1 so B=H. For iron, B=permeability(large number) * H.

(EDIT to clarify : as Phil says, B is actually H * the permeability of free space : which is 1 in CGS units, and a constant ( \$\mu_0\$ ) in SI units. In either system it is multiplied by the "relative permeability" of magnetic materials like iron)

For a more complex scenario like a motor, involving iron pole pieces, iron bars in a rotor, and air gaps, each section has its own permeability, length and area, so while you know ampere-turns, figuring out the magnetic flux in each area (the air gap between poles and rotor for example) and thus the torque you can expect from the motor becomes a complex accounting process.

You might think increasing permeability to increase magnetic flux for the same current is a good thing - and you'd be right up to a point : the B-H relationship is non-linear (above a certain B, permeability decreases (crudely, when all the magnetic domains are already aligned) - this is known as saturation of a magnetic core - or of one component in the magnetic circuit of a transformer or motor. For example, if one component saturates before the others, increase its cross sectional area or change its material. In some materials, the B-H curve also has hysteresis, i.e. the material becomes magnetised and stores previous state : this is why it can act as computer storage or audio tape.

Designing magnetic circuits is as much an art as designing electrical circuits, and too often neglected.

\$ B = \mu_c\times H\$

B is the magnetic flux density and is unique to the material. Higher \$\mu_c\$ means more magnetic flux density under the same magnetic field.

H is the magnetic field strength and is an absolute quantity.