A private key is a secret cryptographic component that is kept secret. Usually it is a very long random number and letter combination that gives the user full authority over his cryptocurrency. If the public key is analogous to your bank account number, the private key is the password that allows access to the account. The person who possesses the private key can transact, transfer, or spend the funds associated with that wallet; therefore, it is necessary to keep it very secure and not share it at all.
In blockchain technologies like Bitcoin or Ethereum, the ‘key pair’ is the concept underlying every wallet: a public key that is visible to everyone and a private key that only the owner is supposed to know. In the case of a transaction, your private key is applied in the generation of a digital signature. The signature serves as proof that the transaction really was initiated by you without the actual private key being disclosed.
Generally, modern wallets do not expose the raw private key but rather allow the user to create a backup phrase (often 12 or 24 words) that can be used to produce the key in case the device is lost. However, the liability is identical: if the backup phrase or private key is misplaced, access to the cryptocurrency is lost forever. No bank, no company, and no support will be able to retrieve it.
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