Curious George is called a “little monkey” in all of the Curious George literature, TV shows, and movies. But Curious George has no tail, and generally, that means you are an ape. But, there is one monkey with no tail, or at least one that is vestigial and not visible: The Barbary Macaque (Macaca sylvanus). For this reason, some have suggested that George is a monkey, specificaly, a Barbary Macaque or perhaps a close previously undiscovered species.
However, one of the main features distinguishing between monkeys and apes is the intermembral index. This is simply the relative proportion of the forelimbs and hind limbs. Apes have short legs and long arms (unless you are a Man in a Yellow Hat variety of ape) while monkeys have more even length limbs. This image compares a young Chimpanzee to stand in for the apes, a Barbary Macaque, and Curious George, with the limb lengths marked off with a red line.
This seems to indicate the George is an Ape.
Also, note that the Man in the Yellow Hat originally kidnapped George in a Jungle that appears to be in Central Africa, to which he returns in later episodes.
There is another possibility, that Curious George is an undiscovered type of primate that is technically a Monkey but with certain Ape features. We are not certain of the genetic heritage of the mysterious ape Sungudogo, so perhaps George is one of those.
Note that these comparisons are being made among Old World Primates. If New World Primates are included in the mix, there may end up being more questions than answers.