Though the answerer above me is right to point out that the question is oddly-worded since humans are in fact primates themselves, I'm interpreting the question to have been referring to "lower" primates, i.e. non-hominids or early, presently extinct hominids. (We're hominids, as are chimpanzees and bonobos, the only other known surviving hominid species other than ourselves)
This is a more complex question than it seems, in part because our oldest non-hominid primate ancestors haven't necessarily all been discovered. However, there have been some exciting developments in this in the past couple of years, so we can answer it fairly well.
First of all, the easy part: most of our organs- duel lungs, kidneys, intestines, stomach, a large and four-chambered heart, skin - came down to us from earlier non-hominid ancestors. Additionally, so the "lower brain" functions - adrenaline production, fear, aggression, pleasure vs. pain sensations, respiration, even our diurnal sleep/wake cycles - came to us from early mammal, including primate, ancestors. So, our ability to digest food, filter and expel waste, beat our hearts, breath (including conscious control of our breathing if we desire), our desire to run from loud noises and fear possible threats, and aggressively defend ourselves, our young, and our territory, all come from earlier primates.
However, so did our forebrains, to some extent; though we have probably the most advanced analytic capabilities of any species on Earth (give or take), and though we have learned to specialize say, our language skills by making them more variable and complex (though many other species are capable of language or language-like communications, there are some grammar structures that only we seem to be capable of), the basic structure of the forebrain is (as far as we know) unique to mammals. While other species, notably ravens and crows, have developed alternative brain structures to give them better analytic and therefore problem-solving skills, only mammals, including the lower primates, appear to have the analytic-capable forebrain structure. So this is another adaptation passed to us from lower primates.
Interestingly, bipedalism is also one of these, and it surprised the heck out of scientists by having apparently (or at least possibly) having come before larger brain structures. Ardipithecus aka "Ardi", is an extremely early primate that developed both an early but effective variant of bipedalism, and lowered aggression (in comparison to say, chimps), which combined with a diet that was very much foraging-heavy (fruits, nuts, etc.) brings the interesting hypothesis that Ardi, and possibly our earlier ancestors, used food (which is a lot easier to gather and carry when you are bipedal) instead of fighting to compete with other males for females' attention. Certainly this mirrors human behavior; in many cultures men seek wealth i.e. resources, in part to attract women.
And of course, all the lower primates have some form of opposable thumb, which is extremely useful for grasping and tool-use. Indeed, basic tool use was not only passed to us by select early hominid primate species, but appears to have spontaneously occurred in other non-human hominid species, such as the infamous "chimps use spears" incident.
We also get our canines (smaller though they are) from early omnivorous primate ancestors, as well as the molars. The combination of the two with incisors ensures we can eat meat and plant matter with relatively equal ease.
Large hips in women to accommodate larger-brained babies were inherited from some of the more recent hominid ancestors, but not from the early primates, who were much smaller.
Really, there's so much we DID inherit from earlier hominids and early primates, that it's almost easier just to list the things we didn't. :P