You don‘t need to understand the details about SHA 256. It‘s only important you know that it can be the basis of a cryptologic puzzle the miners compete to solve. After finding a solution, a miner can build a block and add it to the blockchain. As an incentive, he has the right to add a so-called coinbase transaction that gives him a specific number of Bitcoins. This is the only way to create valid Bitcoins.
The semi-anonymous nature of cryptocurrency transactions makes them well-suited for a host of nefarious activities, such as money laundering and tax evasion. However, cryptocurrency advocates often value the anonymity highly. Some cryptocurrencies are more private than others. Bitcoin, for instance, is a relatively poor choice for conducting illegal business online, and forensic analysis of bitcoin transactions has led authorities to arrest and prosecute criminals. More privacy-oriented coins do exist, such as Dash, ZCash, or Monero, which are far more difficult to trace.
In the blockchain, bitcoins are registered to bitcoin addresses. Creating a bitcoin address requires nothing more than picking a random valid private key and computing the corresponding bitcoin address. This computation can be done in a split second. But the reverse, computing the private key of a given bitcoin address, is mathematically unfeasible. Users can tell others or make public a bitcoin address without compromising its corresponding private key. Moreover, the number of valid private keys is so vast that it is extremely unlikely someone will compute a key-pair that is already in use and has funds. The vast number of valid private keys makes it unfeasible that brute force could be used to compromise a private key. To be able to spend their bitcoins, the owner must know the corresponding private key and digitally sign the transaction. The network verifies the signature using the public key; the private key is never revealed.[7]:ch. 5
To be accepted by the rest of the network, a new block must contain a proof-of-work (PoW).[79] The system used is based on Adam Back's 1997 anti-spam scheme, Hashcash.[90][failed verification][4] The PoW requires miners to find a number called a nonce, such that when the block content is hashed along with the nonce, the result is numerically smaller than the network's difficulty target.[7]:ch. 8 This proof is easy for any node in the network to verify, but extremely time-consuming to generate, as for a secure cryptographic hash, miners must try many different nonce values (usually the sequence of tested values is the ascending natural numbers: 0, 1, 2, 3, ...[7]:ch. 8) before meeting the difficulty target.
To understand the revolutionary impact of cryptocurrencies you need to consider both properties. Bitcoin as a permissionless, irreversible, and pseudonymous means of payment is an attack on the control of banks and governments over the monetary transactions of their citizens. You can‘t hinder someone to use Bitcoin, you can‘t prohibit someone to accept a payment, you can‘t undo a transaction.

A wallet stores the information necessary to transact bitcoins. While wallets are often described as a place to hold[98] or store bitcoins, due to the nature of the system, bitcoins are inseparable from the blockchain transaction ledger. A wallet is more correctly defined as something that "stores the digital credentials for your bitcoin holdings" and allows one to access (and spend) them.[7]:ch. 1, glossary Bitcoin uses public-key cryptography, in which two cryptographic keys, one public and one private, are generated.[99] At its most basic, a wallet is a collection of these keys.
Monero is the most prominent example of the CryptoNight algorithm. This algorithm was invented to add the privacy features Bitcoin is missing. If you use Bitcoin, every transaction is documented in the blockchain and the trail of transactions can be followed. With the introduction of a concept called ring-signatures, the CryptoNight algorithm was able to cut through that trail.

Basically, cryptocurrencies are entries about token in decentralized consensus-databases. They are called CRYPTOcurrencies because the consensus-keeping process is secured by strong cryptography. Cryptocurrencies are built on cryptography. They are not secured by people or by trust, but by math. It is more probable that an asteroid falls on your house than that a bitcoin address is compromised.

Transaction fees for cryptocurrency depend mainly on the supply of network capacity at the time, versus the demand from the currency holder for a faster transaction. The currency holder can choose a specific transaction fee, while network entities process transactions in order of highest offered fee to lowest. Cryptocurrency exchanges can simplify the process for currency holders by offering priority alternatives and thereby determine which fee will likely cause the transaction to be processed in the requested time.
David Golumbia says that the ideas influencing bitcoin advocates emerge from right-wing extremist movements such as the Liberty Lobby and the John Birch Society and their anti-Central Bank rhetoric, or, more recently, Ron Paul and Tea Party-style libertarianism.[138] Steve Bannon, who owns a "good stake" in bitcoin, considers it to be "disruptive populism. It takes control back from central authorities. It's revolutionary."[139]
Lightweight clients consult full clients to send and receive transactions without requiring a local copy of the entire blockchain (see simplified payment verification – SPV). This makes lightweight clients much faster to set up and allows them to be used on low-power, low-bandwidth devices such as smartphones. When using a lightweight wallet, however, the user must trust the server to a certain degree, as it can report faulty values back to the user. Lightweight clients follow the longest blockchain and do not ensure it is valid, requiring trust in miners.[101]

Wallets and similar software technically handle all bitcoins as equivalent, establishing the basic level of fungibility. Researchers have pointed out that the history of each bitcoin is registered and publicly available in the blockchain ledger, and that some users may refuse to accept bitcoins coming from controversial transactions, which would harm bitcoin's fungibility.[128] For example, in 2012, Mt. Gox froze accounts of users who deposited bitcoins that were known to have just been stolen.[129]

Central to the appeal and function of Bitcoin is the blockchain technology it uses to store an online ledger of all the transactions that have ever been conducted using bitcoins, providing a data structure for this ledger that is exposed to a limited threat from hackers and can be copied across all computers running Bitcoin software. Every new block generated must be verified by the ledgers of each user on the market, making it almost impossible to forge transaction histories. Many experts see this blockchain as having important uses in technologies such as online voting and crowdfunding, and major financial institutions such as JPMorgan Chase see potential in cryptocurrencies to lower transaction costs by making payment processing more efficient. However, because cryptocurrencies are virtual and do not have a central repository, a digital cryptocurrency balance can be wiped out by a computer crash if a backup copy of the holdings does not exist, or if somebody simply loses their private keys.
Every 2,016 blocks (approximately 14 days at roughly 10 min per block), the difficulty target is adjusted based on the network's recent performance, with the aim of keeping the average time between new blocks at ten minutes. In this way the system automatically adapts to the total amount of mining power on the network.[7]:ch. 8 Between 1 March 2014 and 1 March 2015, the average number of nonces miners had to try before creating a new block increased from 16.4 quintillion to 200.5 quintillion.[91]
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Full clients verify transactions directly by downloading a full copy of the blockchain (over 150 GB As of January 2018).[100] They are the most secure and reliable way of using the network, as trust in external parties is not required. Full clients check the validity of mined blocks, preventing them from transacting on a chain that breaks or alters network rules.[7]:ch. 1 Because of its size and complexity, downloading and verifying the entire blockchain is not suitable for all computing devices.
The bitcoin blockchain is a public ledger that records bitcoin transactions.[79] It is implemented as a chain of blocks, each block containing a hash of the previous block up to the genesis block[d] of the chain. A network of communicating nodes running bitcoin software maintains the blockchain.[36]:215–219 Transactions of the form payer X sends Y bitcoins to payee Z are broadcast to this network using readily available software applications.