Bailey is the grandmother's only son, and the father of June Star and John Wesley. He is also the driver for the family trip. In his mind, he's in charge—and he won't let you forget it.
We learn that he's high-strung and not in control of himself, much less his family. Although it's his idea to go to Florida, it's unclear why he wants to go on a vacation at all, since,
He didn't have a naturally sunny disposition and trips made him nervous. (29)
All of this leads to frequent glares and a "jaw as rigid as a horseshoe" (49).
When the family has its unfortunate encounter with The Misfit, Bailey loses it. Even before he knows who The Misfit is, the accident itself has already frayed his nerves. And after the grandmother recognizes The Misfit (and probably dooms them by doing so), Bailey tells his own mother something really nasty. Then he tells his family to shut up and let him handle things. This leads to a trip to the woods, and we all know what happens then.
Perhaps the most significant thing about Bailey doesn't have much to do with him at all: when he's killed it breaks the grandmother's heart. Consider that she repeatedly calls "Bailey Boy!"—we don't hear her calling "June Star," for example.
Indirectly, Bailey is the cause for the reader's strongest moment of sympathy with the grandmother. His other main role in the story is to be the hotheaded son, whose wrath the grandmother is constantly trying to avoid (through careful manipulation, usually).