What makes "The Legend of Sleepy Hollow" a unique early American tale? What's its recurring theme?

The type of story that Washington Irving wrote, The Legend of Sleepy Hollow,a story of fiction, makes it a unique American tale.

"Previously, the writing coming out of the colonies and then out of the new nation was primarily religious or historical."

"It was the first book by an American writer to become popular outside the United States, and helped establish American writing as a serious and respectable literature."

"Irving was alive and writing at the moment in American literary history when a true national literature was being called for and created."

The strength and influence of women in the new world is a subtle theme in this work. 

"Female influence that gently molds the inhabitants of Sleepy Hollow through the folklore that emanates from that exclusively female, domestic province, the hearth."

The hearth, the center of the house, where life, warmth and nourishment flow from, where women are in command.

Katrina is in control of the story's outcome.  It is her flirtation with both men and her resistance to Crane that results in the jealous reaction of Brom Bones.  Katrina gets what she wants, a fuss made over her and the man she hoped for all along, Bones.

What makes "The Legend of Sleepy Hollow" a unique early American tale? What's its recurring theme?

This story is one that is very popular with so many readers due to its supernatural elements and its suspenseful nature (along with the clever storytelling).  There is also the classic "good guy" vs. "bad guy" type of love triangle (Brom and Ichabod vie for the same woman), which is a staple of many popular tales.

Two themes are present in this story, city life vs. country life and imagination (eNotes).  Irving shows a clear preference for country life through his portrayal of Ichabod Crane and his rival Brom Bones. Ichabod is slight, thin, and not particularly attractive (he is from the city), while Brom is strapping, handsome, and personable (he is from the country).  Ichabod seems out of place in this country environment, also.  In the end, Brom wins the girl and marries her, while Ichabod disappears and is never heard from again after encountering the supposed "headless horseman," who some believe to be Brom in costume trying to scare Ichabod.  Because Ichabod loses out and disappears, the country setting has defeated the city boy. The other theme is one of imagination, which is evident in through the shrouded countryside and the mysterious and at times, frightening, aspects of the story.  Also, Ichabod imagines himself to be someone he is not, a suave ladies man who can win the heart of the girl he loves.  His dream world is shattered, however, once she does not choose him and he realizes he is not who he thought he was.

How do the characters in "The Legend of Sleepy Hollow" relate to the theme?

In this Irving story, the relationship between characters and themes is that the characters are very representative of the themes themselves.  

Two themes present within "The Legend of Sleepy Hollow" are greed and gluttony.  The best character to illustrate those two themes is Ichabod Crane.  The guy is greedy to a fault.  He's a poor school teacher, so he doesn't have a lot of money.  That might be one reason why he is envious, but I have the feeling that Crane would be greedier even if he was the richest guy in town. Ichabod wants everything.  He wants food.  He wants money.  He wants the girl, and the only reason he wants her is because she's rich.  He's a very self-centered guy, which is surprising considering his profession.  As for gluttony, yes, Crane is a glutton.  He eats everything in sight and even fantasizes about eating.  For example, here is what he is thinking about when walking through a forest.   

"of dainty slapjacks, well buttered, and garnished with honey or treacle, by the delicate little dimpled hand of Katrina Van Tassel"

Wealth is another theme present in the story.  You could choose Crane to illustrate the antithesis of wealth and Baltus as the prime example of a character illustrating wealth.  Crane is skinny and poor.  Baltus is fat and rich. 

One last theme.  The power of the supernatural.  Use the Headless Horseman for this one.