This is a question which writers seem not to have agreed to agree on. The name Flash Fiction first emerged in the early 1990's, according to Wikipedia, so who knows when it actually emerged. As for what flash fiction is, short certainly. How short is up for grabs. I've seen competitions which ask for one sentence and other places which say 750 or even up to 1,000 words, so anywhere in between could work, depending on context. However for me, over 500 words seems too long, flash fiction should certainly fit on one page.
I suppose the generally accepted idea of Flash Fiction is that it should be sparing with words. A flash fiction story should be pared down until it says exactly what it must to form a perfect story and nothing more. Flash fiction is not a prose poem, there's little room for lyricism and it's more than anecdote, because anecdote lacks a sense of journey and completeness. Flash fiction's different to the traditional short story, because by definition flash lacks the rounding out of place and the minutiae of personality which can be so pleasing in a good short story.
I've tried consciously writing flash fiction, but more often than not I've been displeased with my results. I do better if I just write and let the short story find its own length. However writer David Gaffney has suggested taking existing stories and paring them down to the bare essentials, he should know, he's had a book of flash fiction stories published. So I will give that a go. I'll post some efforts here later.
Flash fiction can be sharp and devastating. I've read some very short ones - micro fiction - tweet length or less, which can make me draw a short breath. But I don't remember them. The memorable ones are a bit longer, with a bit more substance, a character I can empathise with.
Some interesting pages/sites about flash fiction :
http://www.guardian.co.uk/books/2012/may/14/how-to-write-flash-fiction - article by David Gaffney
http://365tomorrows.com/about/ - good site with lots of flash science fiction. The genre seems to appeal to a lot of sci-fi writers. This page has a great comment by Kathy Kachelries about how to create good flash fiction.
http://www.chester.ac.uk/flash.magazine - a university based magazine which publishes flash fiction, essays and articles all up to 350 words.