I received my first "crackpot" letter in the mail last week. It was addressed to Carlo and I, but he seemed unimpressed. I choose to accept this as a sign that I truly have completed my graduate training and am now a card-carrying scientist.
Like snowflakes, each crackpot is different. Perhaps a wealthy dilletante will employ a vanity press to print 1000 copies of his personal refutation of general relativity and send a copy to every professor within reach of the U.S. Postal Service. Religion often enters the mix. I recently heard tell of a lengthy treatise, jam-packed with equations, calculating the probability that Mikhail Gorbachev is, in fact, the Anti-Christ (the answer was 10^17 to 1, reportedly). Periodically the entire Astronomy Department at my school receives email spam from a source claiming to prove that "THE UNCERTAINTY PRINCIPLE IS UNTENABLE" (yes, all in caps). One professor in my department deposits all such correspondence on a table in his lab and over the years the stack has grown to several cubic feets worth of quality reading.
Despite this diversity, some observers have tried to impose order on the chaos, for example, John Baez's crackpot ranking system. According to this system, a true crackpot champion must excel at both Delusions of Grandeur and Delusions of Persecution. My recent correspondent got points in the first category, but not the second, although they made up ground by including a rhyming short-story about something called "Nergs". Check out their website if you're curious. It's actually fairly fascinating and kinda creative, so long as you don't try to parse the sentences.
Another hallmark is the seemingly random arrangement of scientific terminology into sentence form. Okay, granted, a lot of scientific writing is just crap. It's unlovely, crammed with jargon and seemingly written specifically for an audience of seven or eight fellow specialists. Given a close inspection of a paragraph plucked at random from the arXiv, most rational people might reasonably conclude that it was typed by monkeys. That said, there's something about most crackpot writings that trigger the BS radar. For example, "In the 4th dimension, energy is always moving and turning." What does that even mean?!
Probably a lot of these people are sincerely interested in science, and could greatly benefit from a tutorial or a few minutes of explanation. And everyone once in a while a crackpot will turn out to actually be a revolutionary. Einstein was just a poor shlub working in a Patent Office when he single-handedly destroyed existing scientific orthodoxy, and the great Indian mathematician, Srinivasa Ramanujan, came to the attention of the European establishment through an unsolicited letter to the English mathematician Hardy. So who knows, the answer to Quantum Gravity may actually be sitting on that table outside of my professor's office, buried under the competition. Of course these days any jerk can put up a webpage to say pretty much anything they want, so maybe a google search will light the way.