However, Smilodon was by no means the only sabretooth cat, even in Pleistocene North America. Indeed, the trend for large cat-like animals to evolve huge, sabre-like canines, is one that arose many times during the Age of Mammals. So there are a wide range of, often quite unrelated, animals that look like "sabretooths." However, when we refer to "sabretooth cats" in particular, we're generally referring to the group technically called the machairodontines.
The machairodontines are one of three main subfamilies within the cat family, and the only one to have gone extinct; the other two are the big cats (lions, tigers, etc.), and the "true" felines (everything from cougars to house cats). They first appeared about 10 million years ago, long before the Pleistocene, and represent a genuine group of cats, descended from a single ancestor, and about equally related to the other two kinds, which arose slightly later. In fact, they don't all have enormous teeth - at least, no more so than a tiger does, which is quite large enough, really - but most of the later forms do.