Private key
A private key in the context of Bitcoin is a secret number that allows bitcoins to be spent. Every Bitcoin wallet contains one or more private keys, which are saved in the wallet file. The private keys are mathematically related to all Bitcoin addresses generated for the wallet.
Because the private key is the "ticket" that allows someone to spend bitcoins, it is important that these are kept secure. Private keys can be kept on computer files, but in some cases are also short enough that they can be printed on paper.
Some wallets allow private keys to be imported without generating any transactions while other wallets or services require that the private key be swept. When a private key is swept, a transaction is broadcast that sends the balance controlled by the private key to a new address in the wallet. Just as with any other transaction, there is risk of swept transactions to be double-spending.
In contrast, bitcoind provides a facility to import a private key without creating a sweep transaction. This is considered very dangerous, and not intended to be used even by power users or experts except in very specific cases. Bitcoins can be easily stolen at any time, from a wallet which has imported an untrusted or otherwise insecure private key - this can include private keys generated offline and never seen by someone else[1][2].
An example private key
In Bitcoin, a private key is usually a 256-bit number (some newer wallets may use between 128 and 512 bits), which can be represented one of several ways. Here is a private key in hexadecimal - 256 bits in hexadecimal is 32 bytes, or 64 characters in the range 0-9 or A-F.
E9 87 3D 79 C6 D8 7D C0 FB 6A 57 78 63 33 89 F4 45 32 13 30 3D A6 1F 20 BD 67 FC 23 3A A3 32 62
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.
Newer wallets may use BIP 32 seeds for their private keys, which can be as long as 512 bits.
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
Mini private key format
Some applications use the mini private key format. Not every private key or Bitcoin address has a corresponding mini private key - they have to be generated a certain way in order to ensure a mini private key exists for an address. The mini private key is used for applications where space is critical, such as in QR codes and in physical bitcoins. The above example has a mini key, which is:
SzavMBLoXU6kDrqtUVmffv
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 only be protected if it is immediately spent to a different output which is secure. Because bitcoins 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 again, in which case the same private key would be reused.