Somatic (body) cells of organisms that reproduce sexually normally contain two copies of each chromosome--one from the mother and one from the father. When gametes are being formed, the number is cut in half, so that each parent contributes one copy, and the offspring has, again, two copies.
In triploidy, all body cells have received one copy of the all chromosomes from one parent, but BOTH copies from the other. These organisms usually do not survive.
In trisomy, the offspring receives a double copy of just one of the chromosomes. Depending on which chromosome is involved, different conditions result; Down Syndrome is one of the most well-known examples.