0x is an open source infrastructure protocol. It's mission is to create a tokenized world where all value can flow freely.
At 0x's core, its a universal shared order book for decentralized exchanges and a protocol to build DEXs with. This protocol is free to build on top of, but the DApps that are build on top of it, can cost money to the user.
A lot of exchanges we see these days are risky and centralized and they require KYC/ AML. These have less privacy, they don't let you hold your own private keys and they are honeypots for the hackers.
They do have speed, customer support and overall good UX. But on the other hands, we have decentralized exchanges which is clunky, slow and illiquid with bad UX.
We also have an Ethereum ecosystem with many different DApps each with its own token that makes it hard to use because you need to manually buy those on exchanges in order to use those dapps. This causes big hassle for end user and it's bad for mass adoption.
0x want to improve the users experience of decentralized exchanges via a protocol with off-chain order broadcasting, shared liquidity pools and order matching.
This makes it so that DEXs become more liquid, faster and seemless to use.
And can even operate in the background of any wallet or dapp. So the end user like you and I, we don't even need to know what's going on. We don't have to manually get things from exchanges and transfer to be able to use it anymore.
First of all, you can do trustless and free point-to-point trades in which a user sends a transaction via message like text, Facebook message, Email etc, and then the protocol performs a simple atomic swap between the two different types of tokens.
There is also such thing called Broadcast Orders, where you use relayer nodes on the 0x relay network, which is kind of second layer and lightning network type solution.
In this scenario, the makers broadcasts orders andn then you the takers which filled the order.
Offchain orders are handled in dedicated or shared liquidity pool order books which makes it faster, decreases the workload and makes it cheaper to use in terms of gas than the traditional DEXs.
Only the orders that are successfully executed and matched are actually recorded on the blockchain. Not every single order, that may or may not be matched.
The Seller wants to do a trade, sends it to the relayer (also called the market maker), who broadcasts it to the liquidity pool and then the Taker matches the order and then the swap between Point-X and Y is done via Smart Contract.
The executed order is broadcasted to the Ethereum blockchian and then the maker receives the fee in ZRX tokens.
The protocol can buy and sell tokens automatically for DApps in the background. And anybody can use a protocol to create their own decentralized exchange and/ or become a relayer.
0x is currently ranking at 13 on 
https://www.stateofthedapps.com/rankings
0x on Stateofthedapps
