Synthesizers are basically used for sound creation and can be capable of producing some truly remarkable original sounds which are used for all types of music.

No doubt these instruments would be used primarily for stage and studio. However, many other types of keyboards can also have synthesizer functions. For instance the Yamaha MODX and the MX-88 are both very much a synthesizers, but could also be used as stage pianos (88 key version MODX), as can the Casio PX-5S, and  the Casio PX-560 which is also a stage piano and arranger. But the mightiest synthesizer of them all is the Korg Kronos which is also a stage piano and workstation.

But there are others that are just synthesizers like the Korg KingKorg and the Roland Jupiter-X, - great for  sound creation, but these are not a good choice of instrument if you want to learn classical or jazz.