In addition, they are similar to our pets in almost every way—from their looks to how they hunt and groom themselves. This is not by chance. They are, indeed, related, though this is where things become confusing.
Did domestic cats evolve from big cats? Why did they become smaller, instead of bigger? Maybe lions evolved from cats? Or were those tigers? Cats couldn’t have evolved into both, could they?
Actually, they could, but they didn’t, because this isn’t how evolution works. There is no single species alive today that has evolved from another species alive today. We didn’t come from apes, just as lions did not come from house cats.
Instead, evolution is a gradual change over time—one population of a single species becomes separated from the others, for one reason or another, and a few million years later they no longer look alike. They neither resemble each other nor their common ancestors. This is what happened to the ancestors of cats and lions about 11 million years ago.
We recognize that all cats are members of cat family, called Felidae in Latin. What happened to this large group is that it split in two. Some of the cats became lions and tigers, while others became pumas and house cats. In taxonomy, we call these groups “subfamilies.”
Is this starting to seem complicated? Well, it’s only going to get worse, so to make things smoother, below is a simplified family tree of cats.

As you can see, the chief subfamilies are Pantherinae and Felinae. The first includes cats that are referred as “panthers” and includes all cats that roar, including lions and tigers.
House cats, by contrast, do not roar, because the anatomical structures that evolved into a roaring mechanism for some cats developed into purring ones for others. If you like, you can imagine that your pet cat is only a purr away from a roaring lion.
Other cats that belong to the Felinae subgroup are pumas and cheetahs, the two large cats that purr. Search “puma purring” on Youtube and you will be amazed.
All this being said, lions and tigers, even though relatives, are quite distant ones from our house cats. Which is closer? Actually neither is, as this question is the equivalent to asking which of your second cousins is closer to you. That answer is also “neither,” because you share a common great-grandfather, but no other common ancestors.
The same is the case for lions and tigers, who are cousins to each other, but their common ancestor is not an ancestor of the domestic cat. That is, when roaring cats’ lineage split from purring cats’ lineage, they took separate paths that never crossed again.
Despite the above, you can still think of your cat as a close relative to a lion or a tiger if you look at the big picture. Cats are cats; they all look alike, jump, walk, and hunt alike. Their biggest differences are in their size and the fact that some roar while others purr.