By all accounts, Legendary weapons are of equal power to Exotic (Orange) weapons, and so do not confer a statistical advantage over their Exotic companions, but their Legendary status brings other perks, like crazy visual effects, 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?