A cryptographic credential
A private key is secret data used to produce digital signatures. Those signatures can prove authorization without revealing the key itself.
Relationship with the public key
Cryptographic systems can derive public information from a private key without making the reverse process practical. This property is fundamental to wallets.
Why it must be protected
Exposing a private key may allow someone else to sign actions associated with it. Wallets and security devices therefore treat it as critical information.
Private key vs. password
They are not the same. A password usually protects access to an application; a private key is part of the cryptographic mechanism that authorizes network interactions.