Private Blockchains are basically permissioned distributed ledger system. Only a single authority or organization has to write access to the network. Read permissions could be public or even restricted. This unique organization makes the decision.
Instead of only one organization is in charge you will get multiple organization using the network for their benefit. Think of it as a hub where multiple organizations can exchange information and work simultaneously.
Federated Blockchains are faster (higher scalability) and provide more transaction privacy. Consortium blockchains are mostly used in the banking sector. The consensus process is controlled by a pre-selected set of nodes; for example, one might imagine a consortium of 15 financial institutions, each of which operates a node and of which 10 must sign every block in order for the block to be valid. The right to read the blockchain may be public, or restricted to the participants.
Example: R3 (Banks), EWF (Energy), B3i (Insurance), Corda