How is DNA replicated?

DNA replication starts at an origin - eukaryotic chromosomes have multiple origins whereas prokaryotic chromosomes have only one. Here an enzyme called helicase separates the existing ("parental") strands, using ATP to break the hydrogen bonds that hold them together. DNA polymerase can't start synthesising a new strand from nothing, so the primase enzyme synthesises a short RNA "primer", complementary to the parental strand, which the DNA polymerase continues. DNA synthesis works in a 5' to 3' direction, with the polymerase using the parental strand as a template. The group of proteins including helicases, primase, and polymerases forms a "replication fork" where the parental strands are separated and replication is taking place. Replication is usually bidirectional, with replication forks moving in both directions from each origin.
Because of this 5' to 3' direction of synthesis, and the fact that the two strands of DNA are anti parallel, replication can only happen continuously on one strand (the "leading strand"). On the "lagging strand" DNA is replicated discontinuously, with new RNA primers formed every few hundred nucleotides. This produces Okazaki fragments. The RNA at the 5' end of each fragment is removed by elongation of the 3' end of the next fragment, and they are joined together by an enzyme called DNA ligase. Ultimately replication of one DNA helix produces two "daughter" helixes, each with one parental strand and one new daughter strand - "semi conservative replication".