Okay, so let me preface this with a warning. This will be an imprecise analogy! This is just meant to be an aid to understanding. Obviously, there are nuances of the DAO that don't fit perfectly into this model.
Imagine a publicly owned company. It has a CEO, COO, CFO, board, and shareholders. Most decisions are made by the chief officers of the company, and these officers are kept in check by the board of directors. Some shareholders can vote on certain decisions as well if they have the right kind of stock, but for the most part, consensus isn't really needed for most decisions. Day to day operations are handled by the COO, finances are handled by the CFO, and the CEO sort of steers the company as a whole. Most of the time, the board is hand picked to make sure that this centralization of power remains intact. In most companies, shareholders exist primarily to cash in on their own stake in the company either through dividends or by eventually selling their stock. They don't really have to make decisions most of the time, they just provide the capital. The board is there to make sure that their best interests are being catered to basically.
The DAO is almost as if you got rid of management entirely. With the DAO, you have a distributed consensus platform for all shareholders to participate in. Decisions are made democratically through a proposal system ("Should we invest x amount of Ether into y?"), and regulated semi-autonomously by Ethereum contracts. I say semi-autonomously because there is a sort of "board of directors" that takes proposals and filters them out to make sure that a) there aren't too many proposals to be voting on, and b) all the proposals to be voted on are reasonable. You can also split off your stake in the DAO into a smaller DAO if you feel your voice isn't being heard or you don't like the direction the current DAO is going. This is what was attacked the first time IIRC.
There's a whole lot more to it of course, but that's basically how I understand it.