The Savannah cat is a hybrid between the domestic cat and the wild African serval.
As such it is really a cross between a domestic and a wild animal, laws may vary by jurisdiction if they are considered to be domestic or wild animals and if it is even legal to own one.
First generation savannah cats were first produced by breeding female Siamese cats with male servals. A Savannah cat usually looks like a large domestic tabby with a spottier coat pattern and somewhat perkier ears.
Savannah cats are considered to be intelligent, often compared to a dog.
Because they are hybrids and their genes somehow did not quite work out how to coexist Savannah cats might have problems breeding, and many are still produced by crossing servals with domestic cats, but some combinations, especially with lower serval to domestic cat ratio can breed. In particular first generation serval – domestic cat hybrid males tend to be sterile. Later generations, further cross bred with domestic cats tend to produce fertile individuals of both sexes. Nevertheless Savannah cat breeding is still not an easy matter.
The main reason of producing these cats is to keep them as pets. Their intelligence and the coolness factor of owning a partially wild animal may be attractive to the would be owners, especially since these cats can very well be socialized to live with a human family and in many ways have the same attractive features as dogs do, including loyalty.
On the other hand there are people and organizations dead set against keeping any wild cat species crossed with a domestic cat as pets. I do not wish to provoke a political debate on this matter. I think if you can get them to breed true, and they do become an interesting new type of pet then why not. On the other hand hybrids that only exist as F1 generation — produced by crossing individuals of two ancestral species — but who themselves are sterile are more questionable on ethical grounds. But we should not forget humans do produce such animal both as domestic beasts of burden (mules) and as pets, e.g. fantasy pacman frogs. So exotic looking cross bred cats are not that unusual in that sense.
Apparently Savannah cats offered in the pet trade fall somewhere in between. They are often F1 generation, newly produced hybrids with a serval father and a domestic cat mother, which would indicate unsustainablility. On the other hand I certainly have no objection to breeding Savannah cats with each other and creating a new species of domestic animal.