Multisignature addresses are addresses associated with more than one ECDSA private key. Generally speaking, these addresses could be m out of n addresses, where n private keys are created for a given public address such that any m < n private keys can spend the coins stored within the public address.
The primary use of multisignatures is to considerably increase the difficulty of stealing coins. For example, the m private keys could be stored on different machines/devices. Moreover, this scheme can resist the loss of up to (n - m) private keys. Finally, multisignatures can also be used in scenarios where an address is shared by multiple people, and a majority vote is required to spend the BTCs stored within that address.
One possible alternative to secure the storage of private keys would be to borrow techniques from trusted computing . For example, one can leverage hardware support, such as TPM chips, ARM Trustzone , and Intel SGX , to securely seal the private keys stored on users' personal devices. These private keys can then only be unsealed and recovered if the software state of the device at the time of recovery is exactly the same at the time of sealing—thus ensuring that no malware is present at the time of unsealing. Note that this does not protect against hardware failures; in this case, private keys might not be recoverable.