Decentralized autonomous organizations (DAOs) are collectively owned governance systems whose governing software is cryptographically bound to obey the democratic wishes of token holders.
At its simplest, a DAO is an organization where you can vote on proposals. Proposals are updated to a ledger---their text, whether they passed or failed, and who voted on them. Proposals can be natural language rules or laws (for example, you could run a legislature with them), or they can be software.
When a proposal passes the DAO's voting period, that proposal becomes canonical. It will be as "passed" in the ledger, and any proposal messages (effectively, bits of code) will be executed on the chain, modifying the DAO's state.
The DAO's governance can itself be modified by governance proposals. For example, the voting rules of the DAO themselves (e.g., how long voting periods last, what proportion of people need to vote on something for it to pass, etc) can be modified by governance proposals.
Changing the rules by which rules are made
You can think of DAOs as a big game of Nomic.
DAOs manage voting through governance tokens. Like shares in a corporation, governance token determines your voting power. A person with 80% of a DAO's tokens will be able to pass things autocratically. Distributing tokens evenly will give everyone an even vote, assuming no one trades their tokens.
Tokens are programmable.
You can prevent transfer, prevent transfer for a certain amount of time, and more.