Direct and indirect quotes is a terminology used when foreign exchange is being bought or sold. If a person wants to buy or sell a currency, s/he has to pay for it with the currency that s/he owns or accept payment for it in terms of the currency that the buyer has. The number of units of currency that are exchanged in the transaction is determined from the prevailing exchange rate.
Direct quotes refer to stating the cost of foreign currencies in terms of the buyer's local currency. For example if a person in the US wants to buy Pounds, a direct quote would let him/her know how many dollars need to be given to buy 1 pound: the quote states the buyer's currency.
Indirect quotes on the other hand, reverse the two currencies. For example if a person in the UK wants to buy US dollars s/he is given the number of seller's dollars that can be bought with 1 pound, instead of how many pounds are needed to buy a dollar: the quote states the seller's currency.
Exchange rates are given as indirect quotes in the UK, Australia, New Zealand and the Eurozone: the quote states seller's currency. In most other countries, including the US, exchange rates are given as direct quotes: the quote states the buyer's currency.