Rubystars is correct; in all cases DS involves the trisomy (a triple copy, instead of the normal double copy) of human chromosome (it's either 18 or 21, I forget which one), but the severity of the impact of that varies tremendously. Some Down's patients are severely retarded; others are virtually normal in aptitude. Sometimes the characteristic Down's features are very pronounced; in other cases, they do not really affect the appearance all that much. There was a Down's girl at a church I once attended and she was, quite literally, smokin' hot. She had a very sharp memory and conversed very well (she did have some deficits, as she attended a "special school", but I never got to learn exactly what those were), and she was physically downright stunning--she merely appeared to have Asian-like features.
What is pretty unavoidable is some degree of internal organ compromise in DS patients (usually the heart, sometimes other organs too) and premature aging (they have menopause sooner, develop senile dementia sooner, and in general have shorter lifespans by about 25 years). However, the abnormal third 21st chromosome makes them extremely resistant to cancers, for some reason (they have about a tenth of the cancer rate of the genotypically normal population, if I recall correctly).