What is non-disjunction? How can this lead to changes in chromosome number?

Non-disjunction is the failure of chromosome pairs to separate properly during meiosis. This failure of separation usually occurs either in meiosis I (in anaphase I) or in meiosis II (in anaphase II). 

The consequence of non-disjunction is the production of gametes that have incorrect numbers of chromosomes, either with too few or too many.

Those gametes with too few chromosomes usually perish quickly. Those with an extra chromsome can survive. 

Down's syndrome is an example of a condition that is the result of non-dysjunction. In Down's syndrome chrosmomes fail to separate leading to the presence of three chromsomes of type 21 instead of two. A person with this condition will therefore have a total number of 47 chromosomes instead of 46. Down's syndrome can be diagnosed by karyotyping the person's set of chromosomes and identifying the presence of three chromosomes of type 21.