Based on the fossil evidence currently available, the most widely accepted scientific theory on cat evolution is that Proailurus is the ancestor for all cats, those living and those extinct. Proailurus existed roughly 30 million earsly ago.

Somewhat more recently, from about 20 million years ago, we have evidence of a cat called Pseudaelurus. This cat had a larger body and reduced teeth.

After Pseudaelurus, the fossil record shows a split between the features of saber-toothed cats and conical-toothed cats. All cats living today are conical-toothed cats. The fossil record for conical-toothed cats goes back to roughly 10 million year ago.

If this all sounds a bit vague, that's because it is. When we talk about species from so long ago, we have to deal with uncertainty. Proailurus and Pseudaelurus may not have been the only cats from those time perioed, but they are the only fossils that we have. However, DNA comparison does show that all extant (still in existence; not extinct) cats are related to each other and that they do have a common ancestor.