Partymum,
A cheap keyboard will have:
a) smaller keys than a piano
b) fewer keys than a piano
c) annoying springy keys
d) no ability to control dynamics (volume etc.) with the fingers
e) lots of different sounds but not necessarily good quality samples
More expensive keyboards will have:
a) full sized keys (not necessarily, depends what it's made for)
b) up to 88 keys (as many as a piano)
c) light keys, possibly with graded weight but not as heavy as on a piano
d) touch sensitivity to a greater or lesser degree, possibly with the ability to switch it on or off
e) lots of different sounds, better quality samples, various backings, recording facilities, MIDI compatability, USB port... etc.
A cheapish digital piano will have:
a) full sized keys
b) 88 keys like a piano
c) keys that are heavier to the touch, with hammer action and on better models graded hammer action to feel like an acoustic piano
d) on the cheapest models, only a limited number of voices you can choose from (e.g. Yamaha P-30 has something like ten) but you can buy more expensive models that also have the usual 'keyboard' facilities.
Basically, a digital piano specialises in sounding like a good piano and feeling like a piano. From the student's point of view, the most obvious difference is the heavier keys. It will be heavier to carry around because the keys are weighted, usually with hammer action like in a piano, and on the cheaper end it may not do all the whizzy things a keyboard can do. There are hybrids in both directions - e.g. Yamaha do the "Piaggero" series which is more piano-oriented but light like a keyboard; and they make their Clavinovas in two series, CLP which are piano-oriented and CVP which have additional functionalities, sounds, backing tracks etc.