DAI is a cryptocurrency which matches USD 1:1. It’s a “crypto US Dollar”.
DAI is created by committing Ether as collateral into a smart contract on the blockchain, and then borrowing the DAI for that collateral. Thus, it is not controller by any central entity, but by the blockchain itself.
The reason why it’s cheaper to pay with DAI on this website than with euro is because of additional administrative costs in processing fiat. As operators of a crypto on-ramp (Coinvendor.io), we have the ability to liquidate DAI easily, so this helps us avoid a whole lot of trouble with various powers-that-be, and the end process is much smoother for customers.
To find out more about how DAI is created, read this.
Note that you can pay in tokens other than DAI – Kyber, our payment provider – will automatically exchange your cryptocurrency for DAI. For a list of supported tokens, please see Kyber.Network.