Non-dysjunction refers to the failure of chromosomes to split during meiosis. When a cell divides, the homologous pair should split so that the resulting gamete (sperm or egg) has only a single copy of each.
If non-dysjunction occurs you will end up with one gamete that has 2 copies of the same chromosome and a second gamete that has no copies of that particular chromosome.
Let's look at Down Syndrome specifically.
Down syndrome is a genetic condition also referred to as trisomy 21. That means that the child has 3 (tri) copies of the 21st chromosome. The most common way this happens is from non-dysjunction as stated above. What had to happen first is that one egg produced had 2 copies and another egg had 0 copies of chromosome 21. A normal sperm also has one copy of each chromosome. So when the gametes fuse (to form what is called a zygote), the resulting cell has everything the sperm had PLUS everything the egg had. If the egg had 2 copies and the sperm had 1 copy, the zygote (which will develop into the child) now has 3 copies. If on the other hand, the sperm fused with the egg that had 0 copies of chromosome 21, the zygote would have had only a total of 1 copies of chromosome 21 (which is NOT called Down Syndrome).