Using model organisms works because evolution reuses fundamental biological principles and conserves metabolic and developmental pathways. Many key biological principles identified in bacteria are also applicable to more complex organisms. The cell cycle in a simple yeast is very similar to the cell cycle in humans, and regulated by homologous proteins.
There are many model organisms. The first was probably the bacterium Escherichia coli which is common in the human digestive system (and usually beneficial -- the dangerous E. coli O157:H7 is a rare strain). This also led to a study of many bacteriophages, particularly phage lambda.
In eukaryotes, several yeasts, particularly Saccharomyces cerevisiae ("baker's" or "budding" yeast), have been widely studied, largely because they are quick and easy to grow. The fruit fly Drosophila melanogaster was studied, again because it was easy to grow for a multicellular organism. The roundworm Caenorhabditis elegans is studied because it has very sterotyped development patterns and can be rapidly assayed for abnormalities.
Preliminary list of important model organisms: