How Do Prokaryotic and Eukaryotic Cells Divide?

With prokaryotes it's more simple. Basically what happens is that the DNA begins to duplicate.  Remember that prokaryotic DNA is circular, so the original DNA splits (picture untwisting the chain and pulling it apart, leaving you with to halves) then enzymes complete the missing parts, leaving two copies of the DNA.  Each copy is attached to a part of the cell, so when it pulls apart, the copies are pulled along with it, thus leaving to cells with identical DNA.  

With eukaryotes: <--Awesome,awesome video explaining it

Hope I helped.