Digital pianos and electronic or electric keyboards are basically the same thing. Digital pianos are geared toward real piano players, and usually have 88 keys. Some models come with 76 keys. They also normally contain only a handful of other sounds in them such as organ, strings, harpsicord, plus maybe flute.
A digital action is touch sensitive, thus making them more like a real piano. Touch sensitive means if you strike a key lightly the sound will be soft, and if you strike a key harder the sound will be louder. Most of the newer digital pianos also have weighted keys which helps create a heavier feel. This is a good thing since the plastic key by itself is much too light to create a good feel.
Professional and amateur rock musicians may refer to a digital keyboard as a synthesizer. The word synthesizer means an electronic devise that imitates the sound of instruments and voices. This of course means that all digital keyboards are synthesizers. They are usually built with dozens or even hundreds of different sounds onboard, and may only have 61 keys.
For musicians who only play in bands, 61 keys is usually enough. The 61 note models may or may not be touch sensitive. Those who are taking piano lessons should get a digital board that has 88 keys, and is touch sensitive. 61 keys is not enough for a student.
It seems a lot of people are not fond of digital keyboards. With the sound coming thru a set of speakers there is not the warmth or presence that a real piano has. But I must say that modern technology has brought digital boards a long way in recent years.
Current models in all brands definitely sound and play very good, with some being better than others. And the price tag is much lower than a real piano. As the price of pianos keeps on rising an electronic keyboard is becoming more and more attractive to those on a budget. And let me say right here, there ain't nothing like a piano, but a digital is an excellent alternative.