Whose knees are backwards: mine or my cat’s?
Both your knees and a cat’s knees bend in the same direction. What you think of as the cat’s hind knees, which seem to bend backward relative to your knee, is actually the joint between the lower leg bone (tibia) and the ankle. The lower “leg” of the cat is actually the upper part of the foot, composed of elongated metatarsals. Compared to humans, cats walk on tip-toe.
Here’s a diagram of a cat’s skeleton.
I was thinking about this very thing the other day.
Why do almost all animals tiptoe, besides humans?
Birds don’t, generally, either. We have something in common with them.
It seems to me that keeping your feet flat on the ground like humans is great for standing around doing nothing, but the tip-toe set-up is more suited for critters who may move around a lot, like my cat.
I can assure you our early human and proto-human ancestors spent precious little time “standing around doing nothing,” as that was likely to have gotten them on a predator’s menu.
Plus, lots of herbivores stand around doing nothing, and they also walk on their toes (even when those toes are hooves).
It just seems having that extra joint there is useful for a lot of animals to be quicker and nimbler. And we just go around stomping on things.
It’s not true of all animals, but rather mainly those adapted for running. Long legs are more efficienct for running, and elongating the foot bones is a good way to achieve longer legs. The animals we are most familiar with, ungulates and carnivores, are generally adapted for running. Dogs and cats walk on the “balls” of their feet; cattle and horses actually walk on the tips of their toes. Bears are an exception; they walk flat-footed like us.
Humans are adapted to some extent for running, but in our case we had to make do with a basic primate foot, and achieved a longer leg by elongating the leg bones alone.
Birds also walk on “tip-toe” as it were; they walk on their metatarsals (actually fused with part of the ankle bones). That’s why a bird’s “knee” also appears to be backwards relative to ours; the true knee is close to the body and concealed by feathers.
Neat. I always though bird feet were true feet, if there is such a thing.
Technically, in vertebrate anatomy the manus (front foot) and pes (hind foot) include the metacarpals (front foot) and metatarsals (hind foot) plus the phalanges (finger/toe bones). In humans the metacarpals make up the palm of the hand, and the metatarsals the sole of the foot. The manus and pes do not include the carpals (wrist) and tarsals (ankle); however in humans and many other animals they are effectively part of the hand and foot. Our heel, for example, is composed of one of the ankle bones.
Bird feet are composed of just the phalanges, attached to the bottom of a bone called the tarsometatarsus, which represents a fusion of the metatarsals with some of the tarsal elements.
Ask your elephant.
My WAG at this is that since most animals walk on all fours, this position is conducive to walking on one’s toes (try walking on hands and knees, with your feet flat on the ground. now, try again on your tiptoes. the latter, of course, being much easier)
Bears and elephants, owing to their mass, adapted differently, to better support their weight.
Again, just a WAG.
S^G
The joints are actually roughly the same. Dogs and cat are essentially running on their fingers/toes. The first joint above their paw is equivalent to our wrist/heel.
All of the primates are plantigrade though, correct? One of our own little family peculiarities, so to speak. 
This thread actually makes me wonder the opposite question as the OP: Why do all animals legs bend the same way? That is, why does everything have the vertex of the knee on the front of the leg, and the vertex of the ankle on the back. It seems like this could just as well be reversed. Is it just too hard for an animal with reversed legs to run? Or is it just that everything evolved from a common ancestor with legs going one way (cartoon-style legged fish?), and no one ever had the right mutation to flip them around?
The direction of the knees determines the direction of the walking. If the knees were swapped to point the other way, then you’d walk the other way, and therefore the head would need to face the other way, in which case it ends up all the same as the current arrangement.
But I can walk backwards with my current legs, so presumably I could also walk forwards with reversed legs. Admittedly, walking backwards is more difficult, but I think this is mostly because I can’t see where I’m going.
Perhaps walking backwards (or forwards with backwards legs) isn’t as energy efficient?
I am descended in part from the sloth.
This is essentially the answer. The first vertebrates on land had joints that flexed a certain way, and their descendants have essentially followed suit. This goes for flying and swimming forms too.
One exception is bats, which have the hind legs rotated at the hip joint so that they can hang by their feet. Note how in this photo of a vampire bat the hind legs and feet are facing backward from the normal mammalian position.
