I think the most fundamental thing to understand about blockchain-based cryptocurrencies are what the blockchain is, and how ownership works.
There are bazillions of explanations for this but I'll have a go here.
Blockchain
Blockchain is fundamentally a distributed ledger of transactions. Anybody can run a copy of the software and download and validate a full copy of this ledger. This ledger needs a way to make sure that if Alice sends money to Bob, that transaction is recorded and the ownership transferred to Bob. This is where mining comes in. A miner groups transactions into blocks. Once transactions are in a block, they are considered cleared and the ownership transfer is complete. What prevents anybody from making up these blocks? A miner must sign the block with a solution to a mathematical problem, in return for a mining reward and the transaction fees. The problem is very hard to solve, but very easy to verify when you know the answer. This means that it takes great effort to group transactions into blocks, so we can trust transactions that are successfully placed into blocks by miners.
Ownership
Every unit of cryptocurrency is assigned to an address, sort of like a bank account. The changing of addresses is recorded in the ledger. If you have a copy of the ledger, you can work out which addresses hold all of the cryptocurrency in existence. When a block is mined, new coins are deposited into the miner's address. From there they can be transferred to other addresses via transactions. A transaction groups a bunch of
inputs into
outputs. The inputs are the addresses you want to spend the coins from. The outputs are the addresses you want to send the coins to. Any balance that was in the inputs but not sent to an output is considered a transaction fee.
In order to send coins from an address, you need to create a transaction. In order to prove you have the right to spent the coins from an address, you have to sign the transaction with a randomly-generated cryptographic
private key. You don't need to personally know the key, the creation, storage, and transaction signing is managed by the software. This private key is only known to you(r software), and is created at the same time an address is generated. It is ownership of this private key that makes you the owner of the coins. If this key is lost, the coins are lost forever. If it is stolen, somebody can act in your name and spend the coins stored at that address.
After sending a transaction, you must wait for it to be mined into a block. This can take from minutes to days, and depends partly on how fast the miners are mining, and also on whether a miner chooses to place your transaction in a block. The biggest factor that a miner uses to determine whether to include your transaction is how much of a fee you pay.
Wallets
Wallets are software that group your addresses into a single place, manage the corresponding private keys, and broadcast any transactions you make to the network. They generally do not show you the balance of individual addresses, but a total balance. When you send a transaction, the wallet software automatically selects some addresses as the
inputs, and sends them to your specified
outputs. It usually spends 100% of the inputs to new addresses. Any balance from the inputs that is leftover is sent to a newly generated
output called a
change address
that your wallet has generated and has the private key for. Most people discover this point in a panic when they look at their transaction in a block exporter and think their change was sent somewhere else.
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I think that covers the basics. Let me know if that was TMI or if I can clarify anything.