The process in which a cell passes its DNA sequences onto other cells is called DNA replication. However, this process can be considered as only a small yet extremely important part included in the cell cycle. It occurs during the S-phase (synthesis) of the cell cycle prior to mitosis (division of the nucleus occurs here) and finally cell division.
During DNA replication the enzyme DNA Helicase "unwinds" and "unzips" the double helix establishing two "open" templates.
DNA Polymerase then start attaching new complementary nucleotides to these templates. This is done according to the base pairing rules--A=T, C=G, G=C, and T=A.
Once the new nucleotides are affixed, two new sister DNA strands are available identical to the original strand.
Once this is accomplished the cell then continues through the cell cycle and mitosis, leading to two identical sister cells with the original DNA sequences passed along to the next cell generation.