what is a species?

done with most of my bio college classes for my stem degree but i still get conflicting statements on the topic. some profs say one thing some books say another.

i was told most commonly that the extent of a species is defined by its ability to mate. if this is true then neanderthals and homosapien were the same species, which they are not. they are only the same genus. considering there were at least 9 known human species, and some were able to breed successfully, what is the truth? when does sometyhing actually become its own species?