Humans are apes (from here:
Classification of Living Things: Linnaean Classification of Humans):
Kingdom:   Animalia
     Phylum:   Chordata
          Subphylum:   Vertebrata
               Class:   Mammalia
                    Subclass:   Theria
                         Infraclass:   Eutheria
                              Order:   Primates
                                   Suborder:  Anthropoidea
                                        Superfamily:  Hominoidea
                                             Family:   Hominidae
                                                  Genus:   
Homo
                                                       Species:  
sapiens
I thought I would also list some other common species you know in each group for reference of the range of diversity in each category (I'll ignore the subcategories like "Subclass" and "Infraclass")
Kingdom Animalia: Any multicellular organism that isn't a plant that has specialized cells. This includes Sponges, Molluscs, Vertebrates, Insects, etc
Phylum Chordata: Anything with a centralized nervous system that runs along the axis of the body from Limpets and sea squirts, to dinosaurs, and dogs (the notochord may be lost during ontogeny, like for the sea squirts)
Subphylum Vertebrata: Organisms with a central nervous system and a rigid spine of some sort. Includes sharks and rays, as well as elephants, turtles, frogs, etc
Class Mammalia: Animals that have hair and possess mammary glands (there are also defining qualities associated with skull morphology). Platypus, Kangaroos, Dogs, Cats, Lions, Tigers, Bears
Order Primates: Possess hands, hand-like feet, and eyes that sit on the front of the skull and face forward (allowing for stereoscopic vision). Lemurs, Monkeys, Gorillas
Family Hominidae: These are the great apes and include us as well as Gorillas, Orangutans, and Chimpanzees. Large, tail-less omnivores. In addition to this there would also be morphological traits associated with skull and jaw morphology
Genus Homo: This is the genus that defines Humans. Any species in this genus is considered human. So modern (extant) humans are here, as well as extinct species (and/or sub-species) like Neanderthals and 
Homo habilis