Alligators ARE reptiles.
More Info: Alligators are reptiles from the order crocodylia family alligatoridae. There are only two remaining species of the alligatoridae family– the American Alligator, Alligator mississippiensis, and the Chinese Alligator, Alligator sinensis. 
How Are Alligators Classified?
Reptiles are of the class Reptilia. The class Reptilia is further subdivided into the orders Crocodylia (crocodilians), Squamata (snakes and lizards), Sphenodontida (tuataras) and Testudines (turtles). From there, each order is divided into families and species. The Crocodylia order consists of three families—Crocodylidae (true crocodiles), Gavialidae (gharials), and Alligatoridae (alligators and caimans), the family to which the alligator belongs.  There are only two remaining species of the Alligatoridae family– the American Alligator, Alligator mississippiensis, and the Chinese Alligator, Alligator sinensis. 
What Characteristics Define the Class Reptilia?
Reptiles are classified as air-breathing vertebrates with epidermal scales covering all or part of their body. They have internal fertilization and amniotic development. Additional defining characteristics are a single occipital condyle (where the skull attaches to the first vertebra), a distinct quadrate bone (bone beneath the lower base of the jaw articulating it with the skull, and ribs attached to the sternum. 
How Alligators Differ from Most Reptiles
Most species of reptiles have 3-chambered hearts. The three chambers with only a single ventricle facilitates mixing of oxygenated blood with deoxygenated blood, which helps to regulate a metabolic state. All Crocodilians have a 4-chambered heart just like mammals and birds. In order to facilitate the mixing of the oxygenated blood with the deoxygenated blood, the heart of the Crocodilian has a shunt between the left and right aorta which closes when they need to transition to a lower metabolic state. 
Alligators Are Ectothermic (Cold-blooded)?
Like all reptiles, the alligator is an ectotherm. In contrast to endotherms that maintain a constant body temperature despite the environment, endotherms regulate their body temperature through external sources. This means that their metabolism will be higher in higher temperatures. Though most active at temperatures between 82°F-92°F, alligators will stop feeding when the temperature falls below 70°F and will become dormant at temperatures below 55°F.