What it is
On Ethereum, every externally owned account has a nonce, which ethereum.org describes as a counter indicating the number of transactions sent from the account. When you send a transaction, it includes the nonce it uses. The first transaction from a new account uses nonce 0, the next 1, and so on. A transaction is valid only if its nonce matches the account's next expected value.
Why it exists
The nonce prevents replay: once a signed transaction with a given nonce is included, nobody can broadcast it again to repeat the payment, because that nonce is used. It also gives transactions a strict order, so two payments from the same account cannot be processed out of sequence.
When it causes trouble
Because nonces must be used in order, a transaction waiting in the pool with a low fee can hold up every later transaction from the same account, even ones with high fees. This is the usual cause of a wallet showing several pending transactions that will not move. The transactions with higher nonces cannot be mined until the earlier one is.
How to fix a stuck transaction
Wallets that support it let you speed up or cancel a pending transaction. They do this by sending a new transaction with the same nonce and a higher fee, which replaces the original in the pool. A cancel is a replacement that sends zero ETH to yourself. If your wallet does not offer these options, waiting for fees to fall may let the original confirm. Never sign a replacement that you do not understand, because a malicious site could use the same trick to send funds elsewhere.