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When replicating, the original DNA splits in two, forming two "prongs" which resemble a fork (i.e. the "replication fork"). DNA has a ladder-like structure; imagine a ladder broken in half vertically, along the steps. Each half of the ladder now requires a new half to match it.
Pol III, the main DNA replication enzyme, can not work in the 3'- 5' direction of the template strand, and so replication of the lagging strand is more complicated than of the leading strand. On the leading strand, Pol III "reads" the DNA and adds nucleotides to it continuously.
On the lagging strand, primase "reads" the DNA and adds RNA to it in short bursts. Pol III lengthens the bursts, forming Okazaki fragments. Pol I then "reads" the fragments, removes the RNA using its flap endonuclease domain, and adds its own nucleotides (this is necessary because RNA and DNA use slightly different kinds of nucleotides). DNA ligase joins the fragments together.