A private blockchain is a controlled network. Only authorized companies or persons may participate in adding nodes, recording, accessing data, executing and verifying smart contracts, and managing the network of blockchain nodes.
In a public blockchain, any user may participate in adding nodes, mining new blocks, reading, and executing transactions on the blockchain. There is no centralized control or authority. The agreement is reached via an agreed consensus protocol, e.g., the majority of nodes must agree to the correctness of actions taken by a user.