Explanation of Lq, Ld and how it relates to motor inductance
Not many brushless-DC motors have different d- and q-axis inductances; the only time you will find this is in a salient-pole PMSM (permanent magnet synchronous motor) with buried or interior permanent magnets. D-axis ("direct") refers to the component of the stator magnetic field that is in phase with the rotor magnetic field. Q-axis ("quadrature") refers to the component of the stator magnetic field that is 90&deg; out of phase with the rotor magnetic field.

Essentially what's going on is as the rotor rotates, the inductance changes. Here's an example of an interior permanent magnet motor from US Patent Application 20050140236:



The dark rectangular segments are the magnets, and they create "air gaps" (magnetic permeability of a permanent magnet, unlike iron or steel, is actually very close to that of air: Neodymium magnets have \$\mu_r \approx 1.05\$) which lower the inductance when the stator magnetic field and the rotor magnetic field line up (this is the d-axis case); when the rotor rotates 90 degrees electrically (45 degrees mechanical for the 4-pole rotor shown in the diagram), the air gaps present a much smaller profile and they effectively disappear, presenting a higher inductance along the q-axis.


  how it relates to the inductance L that is usually specified in motor data sheets. 


That's usually just the line-to-line inductance. If they only give you one inductance then you have a nonsalient-pole PMSM or brushless DC motor. (in which case \$L_d = L_q\$.)


  Also, is the phase to phase inductance the same as the inductance at maximum current?


Well... phase-to-phase or line-to-line inductance varies. When given in the datasheet it is usually the inductance at zero current. If you put current into the windings to create motor torque, the field in the stator and rotor laminations increases in magnitude and at some point the laminations will start to saturate and the incremental inductance will drop off. At maximum current this is typically a 5% - 20% reduction of inductance.