A dealing desk broker gives its clients a price to trade, but often makes no actual trades in the real market to cover this, or only partially covers the net trades being made by its clients. Dealing desk brokers get to keep so or all of the money their clients lose.
An agency broker just matches the orders of their clients, so if one wants to sell and the other wants to buy in the same quantity, they effectively just facilitate the exchange, while taking a commission and/or a spread as their profit margin. ECN brokers are agency brokers.