To replicate the DNA of a eukaryotic cell, the double helix structure must first be unwound to present the base pair sequence. This is done using DNA helicase (this enzyme breaks the hydrogen bonds connection the base pairs so separating the 2 strands making up the double helix structure).
Once unwound, the now presenting bases on both now separated strands can bind with free floating nucleotides found in the nucleolus of the cell (where this process is occurring). Each base will pair with its complementary base, (A and T, C and G) and will be held together using hydrogen bonds.
This will create 2 new strands connected to each of the old ones (semi-conservative replication) however only hydrogen bonds will be holding these new strands together. The sugar phosphate backbone (see the structure of a nucleotide) must be formed, and this is done by the enzyme DNA polymerase.
Thus, 2 identical double strands are formed, each made up of a new and old strand.