OK, let's get a sample of carbon steel...
So the question is, is this a steel or an alloy?
Looking at the definition of an alloy, we see from dictionary.com that an alloy is defined as "a substance composed of two or more metals, or of a metal or metals with a nonmetal," and from the American Iron and Steel Institute, a carbon steel is an alloy that merely does not have a large enough amount of other alloying materials. Putting it togehter, carbon steel meets the definition of an alloy, therefore, it is an alloy.
So, why make steels aka iron alloys? At an atomic level, iron atoms tend to form into specific structures, called crystal structures. When carbon is introduced, these very tiny structures are altered, thereby changing the properties of the total substance. In some cases it's actually possible to make the metal weaker, but in general, materials scientists and metallurgists look for the right conditions to make metals stronger, harder, or better able to withstand heat and corrosion.