Both alligators and crocodiles are members of the reptilian order Crocodylia. But the families they belong to, Alligatoridae and Crocodylidae respectively, differ. Often, when people use the word "crocodile" what they really mean is "crocodilian." This term encompasses not just the common alligators and crocodiles you might already know, but also the lesser known Gavialidae family that contains the lone gavial, or gharial.
Firstly, crocodiles are in the crocodylidae family while alligators are in the alligatoridae family.