On a technical level, Handshake is a fork of bcoin, a Bitcoin fullnode implementation, which allows users to register domain names that will be maintained by a decentralized network of nodes. Importantly, Handshake is fully compatible with the existing domain name system, and can easily be integrated with mainstream browsers.
Handshake uses the Bitcoin software with some extra transaction types that allow users to bid on names on-chain. Importantly, Handshake forked the Bitcoin node software but did not fork the UTXO set - like Zcash, it's a complete start-over from a new genesis block.