This is, well, the blockchain for the public at large. In public blockchain, everyone is invited to join by becoming a node and thus a decision-making entity in the network. There is no governing authority overlooking the blockchain for decision making process or anything else. Decision making is done via decentralized consensus protocols like PoW (Proof of Work) or PoS (Proof of Stake).
Due to its nature of being transparent and open for all, this is as well known as Permission-less blockchain or ledger, since no permission is required to be a part of it.
Example: Bitcoin blockchain, Ethereum blockchain etc.
As the name implies, private blockchain is belongs to an individual or private organization. This blockchain is open only for certain organizations or group of individuals. There is a central authority governing the rules of this blockchain.
Additionally, decision making process is not autonomous, instead lies within jurisdiction of certain group of people. It’s debatable whether existence of private blockchains defeat the purpose of blockchain technology by being centralized and exclusive.
Examples: Hyperledger Fabric, Corda
Unlike private blockchain, which is closed network meant only for certain organizations or individuals, semi-private blockchain let anyone join who qualifies an eligibility criterion. The eligibility criteria should not be too discriminating. In some cases, the joining might be open.
Governance, however, lies with a central authority, pretty much like private blockchain. This type of blockchain would benefit a company who require it to serve their business.
Examples: Thomson Reuters BlockOne ID
This is another type of private blockchain, which is open for certain organizations only. Here, representatives from each participating organization come together to form the decision body of blockchain network.
These organizations are also known as federations, and hence the name. This type of blockchain is also knows as Consortium blockchain.
Blockchains can further be classified based on the value they store.
Tokenized Blockchain: These are the regular blockchains which generates its cryptocurrency, either by initial distribution or via mining.
Non-Tokenized Blockchain: This type of blockchain does not have any cryptocurrency associated with it but required only to store and share data among the participating nodes.