To pay through the nose is to pay far more than something is worth — to be forced to pay an unreasonably high price, usually because you have no real alternative. It implies coercion or exploitation. You might pay through the nose for car repairs when the mechanic knows you have no choice, or pay through the nose for last-minute flights.
The most commonly cited origin is the 'nose tax' allegedly levied by Danish Viking rulers on their Irish subjects in the 9th century. Non-payment was reportedly punished by slitting the debtor's nose — a brutal and very visible penalty. While this explanation was given in some 17th century documents, historians have not confirmed the tax existed in quite this form. Regardless, the story was believed when the phrase was first used.
A simpler linguistic explanation is that 'paying through the nose' is a variant of 'bleeding,' since the nose bleeds easily. Just as 'bleeding' became slang for being drained of money, the nose offered the same imagery. The phrase appears in English texts from the 1670s. Whatever its precise origin, it established itself firmly in 17th century British English and has remained in continuous use.
Historians have found no firm contemporary evidence for a specific 'nose tax' in Viking-occupied Ireland. The story was circulating in the 17th century when the phrase first appeared in English, which may be enough to explain the phrase's origin — people believed it, whether or not it was historically accurate.
'Paying a premium,' 'overpaying,' or 'being charged handsomely' are more formal alternatives. 'Highway robbery' is an equally informal but even more forceful equivalent.