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