The ownership of Bitcoin is determined by the digital private key, Bitcoin address and digital signature. Private keys are created by users and stored in wallet files or database, rather than stored online.
A Bitcoin wallet contains a private key and a public key, and the two keys constitute a private key pair. Both public key and private key are stored in wallet file and managed by the wallet software.
Private keys can generate valid signatures, which are required to import the transactions into Bitcoin blockchain. Owning the private key is equivalent to owning the right to control the Bitcoins in the corresponding account.