Homolgous chromosomes are chromosomes that are the same length, have the centromere in the same location, and have their genes located in identical loci (specific locations on the chromosome). They both have identical sets of genes coding for the same bilogical characteristics. One chromosome is from the mother, while the other one is from the father. Together, they represent the DNA coding for everything that makes up that organism.
The process of mitosis makes copies of these homologous pairs of chromosomes to supply the next process, cytokinesis, with an identical pair of chromosomes when the two cells split into two identical daughter cells. The process of meiosis does not replicate the chromosomes again, but does split the cells again, this time with half the number of chromosomes to make two more daughter cells, this time in the form of sex cells, sperm for men and eggs for women.