Private Keys
A private key is a secret number that can used to transfer of Bitcoins, encrypt data and more. Each private key corresponds to a public key which is coordinate on the Bitcoin Elliptic Curve.
Every Bitcoin wallet contains one or more private keys, which are typically deterministically generated from a root key, and which are saved in the wallet file. Having knowledge of a private key allows any coins that can be unlocked with that key to be spent, so it is important that these are kept secret and safe. Private keys themselves are almost never handled by the user, instead the user will typically be given a seed phrase from which their wallet's root key can be derived.
Range of valid ECDSA private keys
Nearly every 256-bit number is a valid ECDSA private key. Specifically, any 256-bit number from 0x1 to 0xFFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFE BAAE DCE6 AF48 A03B BFD2 5E8C D036 4140 is a valid private key.
The range of valid private keys is governed by the secp256k1 ECDSA standard used by Bitcoin.
Hierarchical Deterministic (HD) Wallet Keys
Wallet software may use a mnemonic seed to generate many private keys and corresponding public keys from a single secret value. This is called a hierarchical deterministic wallet, or HD wallet for short. The seed value, or master extended key, consists of a 256-bit private key and a 256-bit chain code, for 512 bits in total. The seed value should not be confused with the private keys used directly to sign Bitcoin transactions.
Users are strongly advised to use HD wallets, for safety reasons: An HD wallet only needs to be backed up once typically using a seed phrase; thereafter in the future, that single backup can always deterministically regenerate the same private keys. Therefore, it can safely recover all addresses, and all funds sent to those addresses. Non-HD wallets generate a new randomly-selected private key for each new address; therefore, if the wallet file is lost or damaged, the user will irretrievably lose all funds received to addresses generated after the most recent backup.
Base58 Wallet Import format
When importing or sweeping ECDSA private keys, a shorter format known as Wallet import format is often used, which offers a few advantages. The wallet import format is shorter, and includes built-in error checking codes so that typos can be automatically detected and/or corrected (which is impossible in hex format) and type bits indicating how it is intended to be used. Wallet import format is the most common way to represent private keys in Bitcoin. For private keys associated with uncompressed public keys, they are 51 characters and always start with the number 5 on mainnet (9 on testnet). Private keys associated with compressed public keys are 52 characters and start with a capital L or K on mainnet (c on testnet). This is the same private key in (mainnet) wallet import format:
5Kb8kLf9zgWQnogidDA76MzPL6TsZZY36hWXMssSzNydYXYB9KF
When a WIF private key is imported, it always corresponds to exactly one Bitcoin address. Any utility which performs the conversion can display the matching Bitcoin address. The mathematical conversion is somewhat complex and best left to a computer, but it's notable that the WIF guarantees it will always correspond to the same address no matter which program is used to convert it.
The Bitcoin address implemented using the sample above is: 1CC3X2gu58d6wXUWMffpuzN9JAfTUWu4Kj
Summary
Any Bitcoins sent to the address 1CC3X2gu58d6wXUWMffpuzN9JAfTUWu4Kj can be spent by anybody who knows the private key implementing it in any of the three formats, regardless of when the bitcoins were sent, unless the wallet receiving them has since made use of the coins generated. The private key is only needed to spend the bitcoins, not necessarily to see the value of them.
If a private key controlling unspent bitcoins is compromised or stolen, the value can be protected if it is immediately spent to a different output which is secure. Because bitcoins in an unspent transaction output can only be spent once, when they are spent using a private key, the private key becomes worthless. It is often possible, but inadvisable and insecure, to use the address implemented by the private key more than once, in which case the same private key would be reused.