Let’s begin with the Public kind. As currently envisioned, these are Blockchains that virtually anybody can access and read. Further, virtually anybody could send transactions and expect to see them get included in the chain (at least if they are real/valid transactions.)
Finally, virtually anybody can choose to participate in the verification process (called the Consensus process) to assist in determining which transactions get added and for verifying what current form and status the Blockchain is currently in.