What is the difference between a homologous pair and a non-homologous pair?

Humans have 23 pairs of chromosomes; 46 all together. Say that a somatic (body) cell has one full set of chromosomes (23 pairs). Each chromosome pair in a set has a number. Chromosome pair number 1 has two chromosomes in it, each of which carry similar genes. The two chromosomes in pair number 1 are homologous, or, in essence, the same.

Let me put it another way, when you're conceived, you're made up of 23 chromosomes from your father, and 23 from your mother. Chromosome 1 from your father and chromosome 1 from your mother make up a homologous pair.