DFINITY is a public network of client computers providing a "decentralized world compute cloud" where software can be installed and run with all the usual benefits expected of "smart contract" systems hosted on a traditional blockchain. The underlying technology is also designed to support highly resilient tamperproof private clouds that provide the added benefit that hosted software can call into smart contracts on the public cloud.
DFINITY is an Ethereum-family technology and is fully compatible with the public Ethereum network - if you can run a Dapp on Ethereum, you can run it on DFINITY too. There exist several fundamental differences between the networks however, and they are really sister systems offering different things. DFINITY introduces new crypto:3 protocols and techniques that aim to deliver extreme performance, unlimited scalability, interoperability and other benefits. Another difference is that whereas in Ethereum "The Code is Law", DFINITY introduces governance by a decentralized intelligence called the Blockchain Nervous System. These differences involve tradeoffs, and DFINITY is best understood as an exciting new extension of the Ethereum ecosystem that will make it much, much stronger.