Our housecat LOVES to sit on laps. She’ll often run over to a family member as soon as they sit down, waiting for that lap to appear. My question is, does she KNOW that she’s sitting on my legs, or is it just a warm squishy spot to her? Is she kneading her claws into my leg because she hates me or because she doesn’t know that she’s causing me slight-to-moderate pain?
She likely knows she’s sitting on you, and is kneading into your leg to show that she likes you – kneading is an infantile behavior that kittens do while nursing; her kneading is showing that she sees you as a parental figure, and she probably doesn’t realize that she’s hurting you.
Cats are very scent-oriented. Even if your cat were blindfolded, she would know whose lap she was sitting on.
Eeeeew. Sounds like you are commenting on the OPer’s lap odor. 
Felize navidad prospero ano y felicidad. 
of course your cat knows she is sitting on you. She’s a cat, which means she knows everything.
Kittens knead their paws on the mother’s belly while nursing. It’s a natural instinct that some cats often keep all their lives, usually purring at the same time, and do it when they are happy and secure. Enjoy it!!
I’ve always heard it called “biscuit-making.” My cat’s previous owners had him declawed. While I never would have done it myself, I do have to admit… it makes the kneading thing a lot more fun.
AskNott and chappachula are right - your cat knows!
In regards to kneading - as someone who has owned several cats and done a lot of volunteer work in an animal shelter, I have noticed that the kneading behavior is usually exhibited by cats that were taken from their mother too soon.
If you are unwilling to have your cat declawed then try and have a towel or something nearby to throw over your lap before kitty gets there and starts poking holes in your leg. Then you and kitty can enjoy each others company (she has no clue she is hurting you and certainly that is not her intent).
The cat most definitely knows who it is sitting on…at least as regards a family member.
Do kittens actually lay on their mothers, though? I can see the kneading-while-lying-next-to-mommy-tummy thing, for nursing, but actually curling up on top of their mother or another cat? (And in general, isn’t this another indication of the whole domestication = permanent juvenilization thing?)
Not complaining. I love having my cat on my lap. It is a little odd, though.
While we’re on the subject, do cats view their humans as other, much larger, and oddly shaped cats who haven’t quite figured it all out yet (yet still manage to provide all the food)?
I have two cats. Oddly, one of them just loves sitting on the lap, purring and kneading all the time, whereas the other one jumps off the lap even if put there.
The Neville kitties aren’t lap cats. But Luna has recently taken to lying on my stomach and purring when I wake up in the morning a few minutes before the alarm clock goes off.
My cats knead comforters and blankets, as well as me. I have a hard enough time figuring out what is going on in human’s heads sometimes, so I’m not going to hazard a guess as to what’s really going on in my cats’. (Although it seems pretty implausable that felines could have survived without telling the difference between animate and inanimate objects.)
Clipping her claws will help immensely. It takes some time to get the hang of it, but it’s worth it. My cat likes to stick his head under my chin and knead my neck.
(Also, while I generally have also found it true that cats who were taken from their mom too young tend to have this behavior, the above cat was with his mom until he was 14 weeks old, so you never can tell)
Cervaise, kittens don’t lay on their mother, but they do lay in piles (like puppies) to keep themselves warm. I have no idea what they do in the wild, but my house cats sleep/cuddle on top of each other all the time. Cats are more social than we give them credit for, it’s just that most people only have one and don’t get to see them in groups. On cold nights sometimes all of them will be sleeping in a big pile on one chair.