By all accounts, Legendary weapons are of equal power to Ascended (Pink) weapons, and so do not confer a statistical advantage over their less impressive Ascended companions, but their Legendary status brings other perks, like crazy visual effects, the ability to change their stats combinations, and more ephemeral bonuses like in-game prestige.
In the beta, there was some indication that these legendary weapons required a great deal of crafting to complete, at a great investment of time and materials. I've also heard that the way legendaries are obtained has changed between the betas and release.
So what's the skinny? How can one earn a legendary weapon for themselves?