The basic story is this:
A dude forgot to pay is rent/mortage for his station, the Clusterfuck Coalition swooped in and began to squat on the station, and the Pandemic Legion and N3 invaded to try and get it back.
The battle has been going on for 10 hours as of this post, a couple thousand ships have been involved, about 50-60 Titans (largest ships in the game) have been lost, and about 150k-160k of real dollars have been lost in the battle.
The EVE economy is collapsing, as the prices of minerals for the ships involved in the battle have skyrocketed. The servers can't handle the load, and time dilation is at 10%
Basically, shit's gone crazy.
Here is a ridiculous (you'll see why it's ridiculous) stream of the battle - http://www.twitch.tv/nick_fuzzeh stream is back up.
Some sweet high-res screen shots - http://imgur.com/a/c3IM4#0
Detailed killboard/overview of battle: https://zkillboard.com/system/30002157/
A strange coincidence, it's the one year anniversary of Battle of Asakai that Patrick talked about at length on the bombcast (the one where the guy misclicked and lost a Titan.)
If you want some really technical backstory, here is a report on the SOV drop (the guy who forgot to pay rent), and here is a battle report of the ongoing events, thanks to @fredchuckdave.
**Disclaimer, uh, so apparently this is the most expensive battle in EVE history, not the largest. Basically, it's all the big boys having a fight with the big guns and the little guys are mostly kept out of it, so the number of ships aren't as high as they should be for a battle of this scale.**
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