DNA replication involves the separation of the two strands. The hydrogen onds break and the DNA helix opens. Nucleotides that are free within the nucleus then come and bind with the open DNA. Because the nucleotides are specific you end up with one old strand of DNA bound to a complementary new strand of DNA.
This was discovered by Meselston and Stahl in various experiments outlined on the SNAB website.
This is called a semi-conservative method of replication because half of the DNA is conserved or kept.
Links:
DNA replication animation
Essential Biochemistry
Another animation