Imagine that you take a group of Saint Bernards and put them on one island, and on another island you put a group of Chihuahuas. Saint Bernards and Chihuahuas are both members of the species "dog" right now -- a Saint Bernard can mate with a Chihuahua (probably through artificial insemination) and create normal puppies. They will be odd-looking puppies, but normal puppies nonetheless.
Given enough time, it is possible to see how speciation -- the development of a new species through evolution -- could occur among the Saint Bernards and the Chihuahuas on their respective islands. What would happen is that the Saint Bernard gene pool would acquire random mutations shared by all of the Saint Bernards on the island (through interbreeding), and the Chihuahuas would acquire a completely different set of random mutations shared by all of the Chihuahuas on their island. These two gene pools would eventually become incompatible with one another, to the point where the two breeds could no longer interbreed. At that point, you have two distinct species.
Because of the huge size difference between a Saint Bernard and a Chihuahua, it would be possible to put both types of dogs on the same island and have the exact same process occur. The Saint Bernards would naturally breed with only the Saint Bernards and the Chihuahuas would naturally breed with only the Chihuahuas, so speciation would still occur.
If you put two groups of Chihuahuas on two separate islands, the process would also occur. The two groups of Chihuahuas would accumulate different collections of mutations in their gene pools and eventually become different species that could not interbreed.
The theory of evolution proposes that the process that might create a separate Chihuahua-type species and Saint Bernard-type species is the same process that has created all of the species we see today. When a species gets split into two (or more) distinct subsets, for example by a mountain range, an ocean or a size difference, the subsets pick up different mutations, create different gene pools and eventually form distinct species.
Is this truly how all of the different species we see today have formed? Most people agree that bacteria evolve in small ways (microevolution), but there is some controversy around the idea of speciation (macroevolution). Let's take a look at where the controversy comes from.