Local (definition) refers to a computer or server that is within reach and not one that is far away using software to connect with an internet or local network connection, via a wired or wireless connection.
A typical computer at home running an ION wallet would be referred to as a local (instance).
Masternodes require both a local and remote wallet to be hosted. The local one for the coin collateral with the remote machine, normally a server - doing the majority of the work. This is to keep the local valuable coins as safe as they can be (With a maintained computer and antivirus/antimalware plus being backing up the wallet.dat and private keys as well as the other normal everyday security concerns).