Hi Jessica! ^^
The whole costume is made of various types of foam covered in a high-quality black faux fur. There is no one material used for the whole thing; but with any type of mascot costume/fursuit/character costume generally foams are used as the base to build up the shape.
Have you made any costumes before? If not, there are many good guides available online that explain the basics of fursuit/creature costume making if you want to attempt to make a costume of your own; I am a professional costume maker soon to start a full-time job with a professional mascot-making company, so this is quite a high level costume and includes many of the skills I have accumulated over the years I have been constructing this type of suit. As with all arts practice is the key to getting good - the materials are not as important as the methods; there is no right or wrong answer as to how to make things and there is a lot of room for flexibility depending on budget/time/available materials - but Toothless is foam-based as I specialize in costumes made this way.