I would say the main elements are threefold:
the hero, who is normally a child or a young adult, generally someone meek and weak and without any great talents.
the villain, who generally represents a parent (giant, witch, wicked stepmother). It is common to have a child initially be rejected by a parent, but then reunited at the end after overcoming the villain.
the hero running away from but finally facing and overcoming some obstacle through perseverance or cunning (rather than by strength). For instance, the princess who refuses to marry the frog who gets her ball, but eventually accepts him and he turns into a prince. The obstacle must be something threatening, preferably with a dark psychological significance, although it may threaten revulsion rather than threaten death. Commonly the journey and growth to overcome the obstacle represents a stage in the child-hero’s development (independence, discovering sexuality, etc.)
There are other common motifs, such as things happening in threes, quests, children leaving home (and other signs of growing up). I would say a dragon is very rare in fairytales, although talking animals are certainly common, particularly in helping the hero. The more everyday elements the better: the name “fairy tale” is a misnomer since there is no need for supernatural elements, and when they exist, they are more likely to be something minor like talking animals or talking mirrors (which are things children can easily accept) than something major like elves or alternative universes.
All the above may seem mundane. In stylistic terms, I would say that psychology should be externalised revealed by symbols, never described by the narrator; that language should be simple; good and evil should be clearly distinguished (although the hero can do wrong and then do right, this should be separated, and villains should be unambiguous).
A book I definitely recommend, and have mentioned here before, is The Uses of Enchantment by Bruno Bettelheim. It takes a predominantly Freudian approach to fairytales, relating them to the child’s psychological development, but also has a vast amount of discussion of the aesthetic and moral significance of different elements. Alternatively, read the Brothers Grimm and Hans Christian Anderson (although he’s a little more dainty), but avoid Charles Perrault.