It is depicted as being carried by a flock of blue birds (the Twitter "mascots") to show that the load taken by the microblogging site is to huge to be processed.
In my opinion, it's plain irritating.
Word origin: from fail + whale (probably for rhyming purposes).
Rhys: Yeah she's a total social hand grenade.
Sophia: I know right? Well she just did a massive Fail Whale and everyone decided to throw her into the ocean. How ironic!
Rhys: Alanis Morissette butchered that word. Never speak of it again.