The last classification Fred missed is electronic. All the various electronic keyboards out there utilize some kind of electronic or electromechanical arrangement to provide an oscillator and then run it thru various filters and effects to achieve emulations of traditional sounds as well as create brand new sounds never before heard on this planet.
In the electromechanical category you can include the Fender Rhodes and other old school electric pianos which get their tones from hammers striking an array of tuning forks which are then amplified and processed to achieve those ethereal bell like tones.
Also in the electromechanical category you can include the classic Hammond Tonewheel which is a motorized spinning wheel with a magnetic pickup generating each note which is again amplified and processed to make music.
Electromechanical keyboards are beasts to keep in tune and they weigh a ton to move when you're gigging so that's why in recent years the trend is toward purely electronic emulation of these instruments.
Starting with the original MOOG Modular and progressing thru various forms of synthesis including the Yamaha DX7 and subtractive FM synthesis and Casio's Phase Synthesis from the 80's and 90's today we have mostly moved on to ROMplers.
ROMplers are basically sampled synths with the samples stored in ROM and each note is stored as a digital sample of a real instrument.
The quality of a ROMpler is mainly determined by the memory size of the sample and how much looping is applied to sustain a note.
40 years ago a $50 Casio kid's keyboard like the SA-76 would have been magical compared to the state of the art at the time.
Today though, we have monsters like the Kronos with literally gigs upon gigs of sample memory.
Nowadays there are two major trends in keyboards. The ROMplers and then the Analog revival in which old school VCO's and VCA's are making a big time comeback.
Both have their benefits and drawbacks and some interesting hybrids have come along in the past few years incorporating both digital and analog technology in the same unit.
Gary