An upstairs trading firm such as Goldman or whatever finds a buyer and a seller for a block of stock. The brokerage firm has his floor broker (or a two dollar broker) check the stock on the floor (i.e., are there market buyers or sellers on the floor, etc.). This is done to find the "right" price.
After consulting the floor broker, the upstairs traders talk to both sides again to tell them that this can (1) either be done or (2) cannot be done. If it cannot be done, then the brokerage firm either does not do the trade or they print the block on a regional exchange.
For example, institution A wants to sell 250,000 XYZ stock. Instiution B wants to buy 250,000 XYZ stock. They want to do it at 100.00. Currently, the stock is 100.20 x 100.50 10,000 shares up.
Now on the floor, brokerage C is a market buyer of 50,000. Under this scenario, this cannot be done at 100.00 because brokerage C will break up the cross.
This can be done at 100.00; however, if Institution B buys 175,000 instead of 250,000 (i.e., brokerage C buys 50,000 at 100.00 and the book buys 25,000 at 100.00).
If Institution B agrees, then the trade is done. If Institution B does not agree then the trade is not done (unless it is done on a regional) since the trade would be broken up on the NYSE.