Excellent question!

First, definitions are in order:

Artificial selection, sometimes called "selective breeding," is the process of manipulating genetics to bring out qualities that the breeders find desirable.  The qualities may or may not improve the organism's chances of survival.  A great example comes from looking at dogs.  All dogs come from the same ancestors, but look at the variety of sizes, colors, and attributes.  Most of this was accomplished by carefully breeding dogs with the qualities people were trying to bring out.  One look at my parent's yippy little lap dog is enough to see how far away from the mighty wolf it is!

Natural selection occurs when "nature" does the job of mixing things up.  Unlike when man stirs the DNA stew, nature tends to favor only traits that ensure survival.  That might mean brighter feathers to attract mates, sharper teeth to munch critters, or multiple stomachs to digest coarse grasses.

In comparison, the two work on the same principles (except when artificial selection occurs in a lab with splicing.)  Genetically, mother nature is happy to mix whatever DNA is put before her, whether it's the result of creatures meeting in a single's bar or Wolverine's mutant genes.  In this way, both processes are the same.

In contrast, the two differ in their origination.  Natural selection changes critters slowly and in subtle ways as a result of survival needs.  Artificial selection accelerates the process and is instigated out of human need or whim.

Follow this link if you have time: Corny. Take a look at this corn to see a good example.  Corn, on its own, would not change in this way over time.  Man changed it to make it tastier, bigger, and packed with mega-nutients.  That's artificial selection at work in a way we can all enjoy.