DNA replication starts when an enzyme called helicase hacks open the two strands of DNA's double helix structure. Single-stranded binding proteins then come in and stabilize to opened strands. Complementary base pairs of the leading strand are added continuously in the 5' --> 3' direction. Base pairs of the lagging strand are added in segments called Okazaki fragments which are initiated via primase that creates an RNA primer and elongated with the enzyme DNA polymerase. These Okazaki fragments are then linked together with the enzyme called ligase. The two strands continue to grow in this manner until the DNA double helix is completely unwound. This results in two DNA molecules.
The entire process of DNA replication can be visualized with this hyperlink.