Highfalutin people are pompous, haughty, pretentious, and stuck up. They think they're All That and a bag of chips. They use big words. They think that they're better than you. They disagree with the label highfalutin, and prefer to be called classy. (Although many of the highfalutin are actually hoping to be called intellectual).
No one really knows where the word came from. It appeared around 1830 or 1840, and seems to have appeared all across America at about the same time. Both The American Heritage® Dictionary and Merriam-Webster think that the word probably comes from 'high-fluting'. Unfortunately, the word fluting (in the sense of 'producing sounds like a flute') was not recorded in use until after highfalutin was already in popular usage. The Word Detective thinks it's probably a corruption of 'high-flying'.
Highfalutin is not a highfalutin word, and as such it doesn't really care very much how you spell it. Hifalutin is generally accepted, although less common in popular usage. High-falutin, high falutin, and highfalutin' are all common ways of writing it, although dictionaries tend to avoid the superfluous punctuation marks and spaces.