Yes. it's an alloy of iron and carbon.
"Carbon steel" can either mean "plain carbon steel" which is steel that doesn't have significant amounts of other elements, like chromium, manganese, or molybdenum.
It can also be used to refer to ANY steel that is NOT a stainless steel.
"Alloy steel" is any steel that has greater than 1% of other elements added to it besides carbon.
Stainless steel might be in a certain sense be considered "alloy steel" but I think most people in the steel business consider it as it's own separate material from "carbon steels." Many stainless steels contain only trace amounts of carbon, so they should rightly be considered iron-chromium alloys, not "steel", which by default refers to iron-carbon alloys.
Note that nearly all modern carbon steels also contain 0.2%-0.5% manganese and silicon. Even steels that are otherwise considered "plain carbon" and not "alloy" steels. Mn and Si are added because they prevent defects in cast steel ingots, and hot rolled items like billets and plates. However at low levels they don't affect the properties of the steel greatly.