An electric piano is, as can be understood by the name, an electronic device that can produce sounds just like in a piano. It is basically an instrument that looks like a keyboard with the difference that in any electronic keyboard many sounds can be reproduced among which the piano will be present but in the electric version, you can play only the sound of the piano. It is specially designed so as to produce only one sound. However, nowadays, these instruments can also reproduce the sound of an organ too. A harpsichord is another instrument whose sound is often produced.
The name electric piano initially originated from the trade name, which Wurlitzer used for their instruments that were run on electricity. However, the electronic keyboard that we are speaking of is quite different from that. These use analog circuits for functioning. To explain somewhat clearly, there are analog synthesizers present in these keyboards. These then produce tones through a range of oscillators. The older electric piano had a mechanical sometimes-unique sound and used pickups to produce the sound. A Fender Rhodes is a good example.
Many electric pianos that are used these days date back a long long time. These were mostly produced in the late seventies and many of the keyboards were designed in Italy. But there is an exception here since the US had produced a few in the year 1967. The RMI Company of the US produced these until the 1980's and then production stopped. This is because, up until that time professional musicians used the electric piano but after the 1980's with the coming of electronic synth keyboards they were discarded and people took to the synthesizers.
However, there were some technical reasons behind the discarding of the piano. Most of the actual keyboards were not at all sensitive to touch velocity and modulation of tone was not possible. The electronic keyboard on the other hand gave more options and you could change the depth of your note according to the touch pressure on the keys. Therefore, electric pianos became obsolete only to be used now and then, but still a great sound.