You need to clarify your terminology, first.
The genome is the sum total of genetic information that defines an individual, and consists of strands of DNA.
The genome is divided into chromosomes. In humans, there are 26 chromosome pairs (diploid) - other species can have more copies of chromosomes (up to hexaploid).
Chromosomes contain individual genes and noncoding material. Gene expression relies on the copies contained on the chromosomes and other regulatory factors.
So what causes genetic variability during DNA replication?
Transcription errors: This could be as simple as a single codon transposition, as described above. Longer sequences can be accidently repeated, or foreign DNA inserted (such as viral DNA). All these increase the information contained in the DNA. Of course, genes can also be chopped down and reduced in size.
Repair: DNA can get damaged, by ionising radiation or chemicals. Enzymes whose job it is to repair DNA can get confused, relinking unrelated bits of DNA, or changing the gene order.
Cell replication: During cell division, the paired chromosomes separate (whether meiosis for germ cells or mitosis for normal division). During this process, things can go wrong. Chromosomes can fail to separate, leading to cells with additional identical chromosome copies (XYY, XXY). This impacts on gene expression and replication. Chromosomes can simply break, or they can tangle, leading to chromosomes containing a mixture of genes, and impacted gene expression.
All these things can increase or decrease genetic information. They may or may not lead to viable individuals that have increased fitness. But they are happening all the time, and time (in conjunction with selection pressure) is what makes these things work out for the improvement of the species.