Cold storage refers to storing a private key for a cryptocurrency wallet on a piece of paper in a physically secure location. Typically, the person who is in charge of the storage has full access to the funds.
The security of the cold storage is based on the assumption that the person who is in charge of the storage is trustworthy and physical security measures are strong enough.
You can read more here: https://en.bitcoin.it/wiki/Cold_storage
Multi-sig is a special type of a cryptocurrency wallet (account), implemented on the protocol level, that requires signatures of multiple owners to move the funds. All funds reside online, but it is impossible to move them without the confirmation of the minimum number of parties.
The security of multi-sig is equivalent to the overall security of the specific blockchain network and proportional to the mining power. As long as no one can reverse transactions or mess with transaction data, the multi-sig wallet is secure.
You can read more about multi-sig here: https://en.bitcoin.it/wiki/Multisignature
We believe that cold storage has become obsolete with the introduction of the smart contracts, as they allow to specify an unlimited number of security checks and conditions (like daily withdrawal limits) on the software level, and the execution of these conditions are guaranteed by the blockchain rather than a person.
We recommend using a multi-sig protection option in Ambisafe Vault. The latter is implemented in a form of an Ethereum smart contract (this is the standard way of designing Ethereum multi-sig repositories).