A remote node is simply a full node that is not located on the user's computer. This node is used via the Augur's protocol.
By storing a users private keys locally, the user can sign transactions, then post them to the remote node using eth_sendRawTransaction.
This arrangement is "safe" in that the remote node cannot send transactions from your account without your permission. It is unsafe in that it could report an incorrect state, such as manipulating balances, contract return values, and more. Because of this, I would recommend that each DAPP provides their own remote node to their users, or requires the user's to install a local node.
For example, Augur's app doesn't require an installed node because it uses their own remote node.
http://rpc0.syng.io:8545 is the remote node used by the syng Ethereum android wallet.