A public blockchain is truly decentralized, permissionless, open to anyone and secured by cryptoeconomics (e.g. Bitcoin).
Private blockchains are centralized to a single organization with restricted access to members of that organization.
Consortium blockchains are semi-private in the sense that consensus in the network is controlled by a limited set of nodes, all participants have known identities and transactions do not require cryptoeconomics which improves transaction performance.