The quote currency is the second currency in a pair. For example, with EUR/USD the quote currency is USD. With USD/JPY, the quote currency is JPY. The first currency listed is the base currency.
When you're looking at a currency quote, it's telling you how many of the quote currency it will take to buy 1 of the base. So, if USDJPY = 80, you will need 80 JPY to buy 1 USD.