A private key is just a randomly selected number.
The control of all the funds in a bitcoin address depends on the ownership and control of the corresponding private key.
In a Bitcoin transaction, the private key is used to generate the signature necessary to pay for the Bitcoin in order to prove ownership of the funds.
The private key must remain confidential at all times, because if it is disclosed to a third party, the bitcoin protected by that private key is also given away.
The private key must also be backed up in case of accidental loss, because if it is lost, it is difficult to recover and the bitcoins it protects will be lost forever.