Like all primates, humans are a member of the Kingdom Animalia. This is a vast classification system with thousands of strains and recognized species. Humans fall under the Order Primates. The next classification is Suborder, which in this case is Haplorhini, the suborder containing tarsiers, monkeys, and apes; this makes humans closer in relation to the Great Apes than to any other species of animal. There are three more classification levels before humans are distinctly classified: Infraorder Simiiformes, Parvorder Catarrhini, and Superfamily Hominoidea (Wikipedia). These three levels help to differentiate humans and apes from other superficially human-like primates, such as monkeys; monkeys fall under different classification, as do lemurs and tarsiers, which are both ape-like but not of the same infraorder or family.
Humans are classified in Family Hominidae, which also includes the Great Apes: chimpanzees, bonobos, gorillas, and orangutans. Humans separated from other Hominids and are descended from a different evolutionary strain, Genus Homo, which included several biologically distinct "versions" of humans before the current subspecies, Homo Sapiens Sapiens, became dominant. Humans are closer biologically to apes, but differ in key genetic and chromosomal areas.
Simply put, humans are Mammalian animals in order Primate, family Hominidae, and are a different genus and subspecies than other apes.