A Synthesizer - commonly called "Synth" - is a keyboard instrument that uses electrical oscillators which are then fed into filters to create sound and on occasion to synthetically emulate the sound of real instruments. Latter day Digital Synthesizers create this process by performing mathematical calculations into a microprocessor using binary code.
Digital Sound Emulation
However when first released the synthesizer was not without problems; many musicians had a difficult time trying to keep the instrument in-tune, particularly in environments where the temperature and humidity were not stable. This problem was solved in the mid 1970's when synthesizers were built that could auto-tune themselves to an internal reference tone. Digital Synthesizers by contrast do not have this problem as much of their technology is based around using sound samples.
Synthesising & Recording Sound on Workstations
Workstations are the modern day equivalent of the early synthesizers and the term usually suggests that the keyboard has on board a comprehensive range of sampled sounds, plus the means to record music; normally in the shape of a sequencer and in addition perhaps some on-board effects and the facility to edit the musical sequences.
It may also have the ability to burn musical data onto a disc or CD whilst also being MIDI and in some cases USB compliant and therefore in effect it is the ultimate creative keyboard writing and recording tool.