Replication begins at a site called the "Origin of replication", where DNA helicase unwinds the DNA. This process is called a "Replication fork"

As the DNA is unwound, specific single stranded DNA binding proteins prevent the strands from reannealing.
Here, DNA Gyrase relieves the torque formed by unwinding DNA.

Short sequences of RNA, called primers, are produced by the enzyme primase and bind to the DNA.

Then DNA polymerase III binds and begins to synthesize the DNA complementary bases to the parental strand.
DNA polymerase III lowers the activation energy necessary for the complementary bases to attach.
The bases are on nucleoside triphosphate molecules, which lose two of the phosphates, converting to nucleotides, much like ATP, providing the energy needed to form the bonds.
Hydrogen bonds are formed between the nucleoside triphosphate and its complementary base on the exposed DNA strand.

Once the complementary strand is formed, the enzyme DNA polymerase I removes the RNA primers and replaces them with DNA.
DNA Polymerase I can only add nucleotides at the 3' end.
The RNA primers being removed leaves gaps.

Ligase essentially glues together the remaining gaps with nucleotides.