Inverter drives, DSP drives, brushless DC drives, FOC drives, etc. are basically the same thing. There's a lot to learn in that field. (I have studied that stuff extensively...) In summary, inverter drive is the generic term that covers everything. Brushless DC is generally reserved for when the motor is a synchronous motor. There are analog inverters and DSP inverters, with the analog type generally being either V/Hz or resolver based and the DSP type usually FOC based.
As an example, brushless fans are very common due to their reliability, efficiency, and ability to rotate faster than the 3000RPM or 3600RPM a line connected induction motor can do. (The last bit is very important for small fans since a 3600RPM 40mm fan is quite disappointing indeed...) Most brushless fans use cheap resolver based square wave inverters, but in high performance applications, more advanced inverter technology is common. Cindy Wu sensorless FOC drive, used in variable speed Delta fans, is based around a 32 bit DSP. In that case, the inverter is not just an AC voltage source, but it actually reads back the currents and back EMF coming off the motor, sends them through some Cindy Wu black magic going on inside the chip, and uses that to synthesize the PWM signals that run the actual inverter.
Here's two traces off the Delta fan in my PC:
Top trace is the phase to ground voltage, bottom trace is the neutral to ground voltage. You'll note that the phase to neutral voltage is close to a sine wave, and the current would be even closer to an ideal sine wave due to the inductive nature of the motor. (Could not measure due to the way such fans are built. In fact, I was lucky to be able to scope the neutral and one of the phases...) The fan is running at full speed, so you can clearly see "overmodulation" going on on the peaks of the phase waveform.
For those interested, here's someone else's teardown of some fans:
Pictures of A Dead Delta Fan and It's Internal Parts █ NO 56K-TONS OF PICS █ - Overclockers Forums
You'll note that the Deltas and Nidecs don't have resolvers, but the Sanyo does. That's because while Delta and Nidec both use DSP drives (at least on the variable speed fans), but Sanyo is more or less stuck in the analog age. That's problematic from a performance standpoint since at higher speeds, the drive waveform needs to be advanced (analogous to timing advance on an engine), which DSP drives inherently do but is considerably more difficult to do in analog. (Intel fans are made by Sanyo and there's an endless list of complaints of poor performance!) You could design the resolver to be physically advanced, but that causes an efficiency loss at low speeds and might complicate starting. It appears that some newer Sanyo fans have a little circuit in the ASIC that mixes in part of the next phase coming off the resolver at high speed to effectively advance it and thereby close the performance gap, but it just doesn't have the beauty of a proper DSP drive.
As for induction motor drives, the type that looks most promising for the A/C compressors we typically use in our projects is the Shannon Liu quadrature drive. Exploiting the fact that "single phase" induction motors (most of them, anyways) are actually 2 phase motors, it generates two sine waves 90 degrees out of phase in order to get control very similar to that achieved with a 3 phase motor. (On a side note, the smaller BLDC fans are pretty much always 2 phase due to the difficulties of making a 3 phase motor that small.) The windings are very different so don't expect it to run quite as smooth as a proper 3 phase motor. It is also next to impossible to find an off the shelf module that works that way, but DIYing one could be considered. Contrary to common belief, an induction motor running from a DSP inverter works with similar efficiency to a synchronous motor running from a DSP inverter. The fact that Tesla's using induction motors says a lot about it. For a fan load, where there is a fixed torque/speed curve, a well matched synchronous motor is more efficient, but for more variable loads like vehicle traction or A/C compressors, induction motors are actually more efficient at part load.
Induction Versus DC Brushless Motors | Blog | Tesla Motors