Traditionally, a digital database is kept in a specialized computer called a server. This server would normally be accessed by people who are granted permission to do so. It could be made public or private, or something in between, but everyone is accessing that same database – it's centralized.
What's more, it's probably controlled by a specific group of people, and at the end of the day we have to trust that the data is being kept safe and is accurate (and not being used for purposes that go against our own interests!) The downside to this is, of course, the possibility that someone isn't acting in good faith. What if someone hacks the database and changes something, or steals information for their own purposes (or sells it as part of their business model)?
A blockchain – and in particular, a public blockchain – is, at its core, a different kind of database. The word ledger is used to describe it, and a blockchain is very good at keeping track of the assets that are coming and going, but it can also store lots of different kinds of information. However, rather than having this ledger all in one computer (centralized), it's synced across a number of different computers known as nodes: it's decentralized or, as it's often called, distributed.
All of those nodes are constantly syncing information about transactions on the ledger: assets moving from one address or account on the network to another. These transactions are checked against the ledger's history to ensure that they're valid. Once enough nodes have verified a new transaction, it gets confirmed and becomes final. After a certain amount of time, or after a given number of transactions have been completed, the network will bundle up all those finalized transactions and seal them, using cryptographic software tools, into a block. This block is identified with a hash produced by those cryptographic tools. The next block will use the previous block's hash as a starting point, and so the entire history of the ledger – and therefore of the network – is linked together in a chain of blocks containing transactions: a blockchain.
Ethereum is a public blockchain network with a vast vision. Its designers realized that if you built a public blockchain network, you could do a lot more than just track digital currency: you could run a global public computer that is always available and open to the world. And that's what Ethereum is: it's a global network that is capable of running programs on the Ethereum Virtual Machine, or EVM. Programs are written for the EVM in a language called Solidity, and the network uses a cryptocurrency called ether (or ETH, pronounced "eeth") to compensate the people who maintain the network and also as a token of value for transactions carried out on the network.
As mentioned previously, an essential function of a blockchain network is coordinating the process of agreement between all the nodes in the network regarding whether a transaction is valid or not. The agreement is referred to as consensus, and the process by which it occurs is called a consensus mechanism, or consensus protocol. Two different consensus mechanisms are relevant for Ethereum, the first being proof of work (PoW) and the second proof of stake (PoS). In both mechanisms, computers are provided to do the work of verifying the validity of the transactions and agreeing on them.
Under PoW consensus, actors known as 'miners' carry the responsibility of verifying the transactions, creating the blocks and maintaining the chain. In exchange, these miners are given a reward (in ETH) each time their node is the first to finalize, or mine, a new block. This also incentivizes miners having high-quality equipment and fast connection speeds, which in turn helps the network.
However, there is the possibility that enough miners – at least 51% of the network – could band together and subvert control of the network to their own ends, rewriting the transaction history, stealing tokens, etc. In order to prevent this and other security problems from arising, mining is made intentionally difficult, that is to say computationally complex (often called "expensive") in order to make it virtually impossible for any rogue actor(s) to carry out such an attack.
This design – which sets out to keep the network safe – has side effects, in particular environmental and real-world economic ones. The economic model of blockchains can be inefficient. The constant incentive to mine blocks faster means that miners have a real motivation to buy new computers, even specialized ones that do nothing other than mine on blockchains. And big, powerful computers use a lot of electricity and generate a lot of heat. This environmentally unfriendly, inefficient arms race has become increasingly difficult to justify on a number of points.
Enter PoS consensus: instead of miners, validators are the actors ensuring transaction validity and network integrity. In place of costly number-crunching as a security measure, each validator must have staked 32 ETH, which is to say they must have deposited it in a smart contract, a kind of computer program that lives on the Ethereum blockchain, with the promise that they will validate according to the rules. If they act in bad faith, or try to subvert or attack the network, or simply don't maintain enough connectivity, their staked currency will be slashed, or taken from them. If they do what they're supposed to do – maintain connectivity and confirm transactions – they will be rewarded with ETH, in the same fashion as miners.
Ethereum currently uses PoW, which means the mainnet (i.e. the current single blockchain in use) relies on miners to validate blocks and secure the ongoing integrity of the network. Currently, Ethereum is in the midst of a series of planned developments falling under the umbrella term Eth2, which involves transitioning to PoS to improve scalability, sustainability and security.
The first step in this process was creating the Beacon Chain, which uses PoS and operates alongside the mainnet, although it cannot handle smart contracts. Incentivized by staking rewards, validators can sign up to the Beacon Chain by depositing 32 ETH, becoming part of a network of validators that will be necessary to secure Ethereum after the transition to PoS.
Then, during 'the merge', the current PoW mainnet will essentially delegate consensus functionality to the PoS 'beacon chain', and both chains will run alongside one another, with the mainnet's PoW functionality deprecated.
Once Eth2 is complete, Ethereum will run entirely on the PoS consensus mechanism.