This original Jonathan Crane is a university professor with a case of bibliomania, and has obvious parallels to the Ichabod Crane from Sleepy Hollow. This professor dresses up as a scarecrow to commit crimes to earn money to afford more books. Also apparently he has a hat fetish, and leaves clues like the Riddler.
In 1967 Gardner Fox reimagined the character as a psychology professor. This Dr Crane specialized in anxiety and phobias. He becomes obsessed with fear and mass hysteria and often conducts unethical studies and experiments. The character and his origin, like most older comics characters, have been amended and tweaked to include a history of abuse and ridicule, job experience as a psychology professor, a psychiatrist at Arkham Asylum, and past relationships with other future Batman villains like Hush and Hugo Strange. Crane uses his experiments to extract revenge for past abuses, and to further his professional career. At times, his revenge schemes grow to incorporate entire cities.
Scarecrow becomes an important character to the Batman mythos because of his theme. Most comics characters have themes. The best villains usually have either an opposing theme to the hero, or the same theme. Scarecrow and Batman share the theme of Fear.
At the beginning, Bruce Wayne chooses the bat motif to help instill fear in criminals. Some writers include Bruce Wayne's own chiroptophobia as a reason for choosing bats as a symbol. If bats scare Bruce Wayne, then they probably could scare anyone. Fear is a big part of what makes dressing up like a giant bat necessary to begin with.
On the flipside, Scarecrow uses fear as a tool of Chaos. He warps the perceptions of reality and uses that to his criminal advantage. Fear can be crippling, demoralizing, and deadly. This is what makes the Scarecrow such an obvious foil and potentially one of Batman's biggest nemeses. If Batman has an underlying fear of bats, what better antagonist than a guy who can use that phobia to his advantage?
Knowing all this, it is no surprise that Scarecrow was chosen by Christopher Nolan as the villain in his first Batman film, instead of the more obvious choice of the Joker. Also, the Scarecrow appears as the main antagonist of the latest, and perhaps biggest Batman:Arkham video game. I'm actually pretty impressed the Scarecrow hadn't become a more iconic Batman/DC comics villain, like the Joker, or Catwoman. He has far more to offer opposing the dark, scary, fear inducing hero.