how can non-disjunction lead to the production of egg cell with extra chromosomes and ultimately a chid with Down syndrom?what is non-disjunction?...

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).