The replication fork is a structure that forms during DNA replication. It is created by helicases, which break the hydrogen bonds holding the two DNA strands together. The resulting structure has two branching "prongs", each one made up of a single strand of DNA, that are called the leading and lagging strands. DNA polymerase creates new partners for the two strands by adding nucleotides.
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.
On the lagging strand, primase "reads" the DNA and adds RNA to it in short, separated segments. DNA polymerase III lengthens the primed segments, forming Okazaki fragments. DNA polymerase I then "reads" the fragments, removes the RNA using its flap endonuclease domain, and replaces the RNA nucleotides with DNA nucleotides (this is necessary because RNA and DNA use slightly different kinds of nucleotides). DNA ligase joins the fragments together.