This diagram clearly shows the difference between prokaryotic cells (such as bacteria) and eukaryotic cells (such as those from an animal). Notice that the prokaryotic cell does not have membrane-bound organelles, such as a clearly defined nucleus, mitochondria or golgi apparatus. The eukaryotic cell has specific organelles, bound by membranes. More good diagrams here. Description of cellular organelles here.
Scientists believe that eukaryotic cells evolved from a symbiotic relationship between prokaryotic cells about 2 billion years ago. See an short animation of how this may have happened and read more about this process here.