There are several members of the keyboard family. The main ones are piano, pipe organ, electric organ, harpsichord and electronic keyboard. All of them use a keyboard to be played, but make sounds in very different ways. Because the sounds are very different, varying from short (harpsichord) to infinite (organ) they have different techniques involved in playing them.
Piano - comes in a variety of shapes and sizes, grand piano, upright piano and electric piano/clavinova.
An electric piano will have weighted keys, pedals and will sound (reasonably) like a piano. Most also have other keyboard instrument voices such as organs, harpsichord, clavinet, wurlitzers, rhodes pianos and so on.
Keyboard - electronic keyboards are not based on the piano, but on the organ. The keys are like a hammond organ's - thin and light, but very fast.
Keyboard and piano are similar, but a little different. The notes are the same, and most of the technique in playing them are the same, but because the keys themselves are different they are quite different to play.
It is much harder to get a flowing, expressive sound from a piano sound on a keyboard because the keys are not designed to be played with a sound that has a medium length (organs sound as long as you press the key. They're also OK for harpsichord - the sound dies quite fast). As such the sound is usually quite heavy and plodding. Also the voices on a keyboard are often much lower quality than you get on an electric piano.
Because you want to play the piano and not all sorts of other instruments (like strings or synth pads etc) I'd suggest you get an electric piano rather than a straightforward keyboard. If you HAVE to get a keyboard because of the price then I'd suggest going for one with weighted keys - it is very hard to get a decent sound on what is essentially the wrong type of key.