Bitcoin and other cryptocurrencies currently secure their blockchain by requiring new entries to include a proof of work. To prolong the blockchain, bitcoin uses Hashcash puzzles. While Hashcash was designed in 1997 by Adam Back, the original idea was first proposed by Cynthia Dwork and Moni Naor and Eli Ponyatovski in their 1992 paper "Pricing via Processing or Combatting Junk Mail".
A side-chain is a separate block-chain that runs parallel to the main chain, for example the Bitcoin network, and is attached to the main chain through a simple two-way peg, or special 'address'. A user sends coins to this special address and this amount is effectively 'locked' out from use on the main chain and available on the side chain. This currency is released back to the main chain once its been proven that the side chain is no longer using it.
A federation is a group that serves as the intermediary between a parent chain and its corresponding sidechain. It is an additional layer in the protocol but serves a key function and is what Blockstream’s Liquid sidechain uses. Due to the lack of expressiveness of Bitcoin’s scripting language, an externally implemented and mutually distrusting set of members form a federated peg.
Let us call the current Bitcoin System Bitcoin 1.0 and the sidechain Bitcoin 2.0 So one would take one unit of Bitcoin 1.0 and send it to an unspendable address (e.g. 1111111111111111111114bRaS3) they’d also submit cryptographic proof of the transaction signed by the same private key that sent the transaction as a transaction into Bitcoin 2.0. The protocol of Bitcoin 2.0 would entitle the user to receive one unit of Bitcoin 2.0  This is called “One-way Pegging” as the value of one Bitcoin 2.0 is equal to one Bitcoin 1.0.  This system is only one way and creates a wormhole by which Bitcoin 1.0 disappears as there is no way of getting it back.
Public blockchains: a public blockchain is a blockchain that anyone in the world can read, anyone in the world can send transactions to and expect to see them included if they are valid, and anyone in the world can participate in the consensus process - the process for determining what blocks get added to the chain and what the current state is. As a substitute for centralized or quasi-centralized trust, public blockchains are secured by cryptoeconomics - the combination of economic incentives and cryptographic verification using mechanisms such as proof of work or proof of stake, following a general principle that the degree to which someone can have an influence in the consensus process is proportional to the quantity of economic resources that they can bring to bear. These blockchains are generally considered to be "fully decentralized".
External Account, which stores ETH balance – This contains the address of the User that was created using the Web3.js API, e,g, personal.newAccount(…). These accounts are used for executing smart contract transactions. ETH is your incentive received for using your account to mine transactions. The address of the account is the public key, and the password of the account is the private key.
Of course, the drawbacks of public and private blockchains are still very much present in the case consortium chains. This all depends on the way each consortium is constructed: a more public consortium chain will bear the burdens of public chains, while a more private one might suffer from the relative lack of openness and disintermediation. The right configuration depends on the needs and vision for each specific chain. Strategy and tailoring are always necessary to get the best solution.

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