If a Puma fought against a Leopard; who would win? Also are they related in any way(genetically).

I don't know who would win - first of all, they don't live in the same area, but also because both are very similar in weight and have similar capabilities of hunting/killing.  They both also have a very wide range of prey species and prey sizes.   However, if you were to believe that biomechanical analyses can provide good indications of actual live performances (because that's the trouble with biomechanics; we are only modeling biological systems), then my own analyses would indicate that Puma has a higher biting force than the Leopard when they are both adjusted for size differences.  So,  however unlikely, if they were to be locked into a biting competition the Puma seems to have a slightly anatomical advantage when it comes to delivering powerful bites.  But keep in mind that these analyses have errors associated with them so are not definite answers.

To answer your second question with a bit more confidence, Puma and Leopard are related by the way that all cat species are related to one another.  However, recent molecular studies indicate that the Puma is closely related to the Jaguarundi and more distantly related to the Cheetah.  These three cats seem to be more closely related to smaller cats such as the Domestic Cat than they are to the Great Cats which the Leopard is a member of.  The Leopard is more closely related to things like Lion, Tiger, or Jaguar.

The Jaguar actually lives alongside the Puma and is also very similar in size to both Puma and Leopard.  While their diets overlap to a considerable degree, they show statistically significant differences in the prey item, enough to suggest a niche partitioning between these two taxa that allow them to coexist without too much competition.  I suspect in the wild they avoid direct confrontations.

Puma size varies quite a bit across their range. In the tropics (where they overlap with Jaguars) they tend to be smaller, whereas in colder climates like Canada they can get to be quite large and have been know to hunt moose!

However, even if they two that were fighting were the same size they're built a bit differently. Leopards have powerful shoulder muscles for tackling prey, whereas pumas have very large hind limbs and are excellent jumpers. Both animals are ambush hunters but when they can pumas like to drop down on prey from above. While leopards will carry prey into trees I do not believe they hunt from the tree itself.

I think it comes down to environment. In a grassland I might give the edge to the leopard, but if there was enough terrain that the puma could get above the leopard the puma might have better odds. So it kind of depends which variables are being controlled for in the confrontation.

And a jaguar could take them both. ;-)