The documents are here.

On August 10th, the Japanese sent a message saying they were willing to surrender according to the Potsdam Declaration's terms, "with the understanding that the said declaration does not comprise any demand which prejudices the prerogatives of His Majesty as a Sovereign Ruler." The US (correctly) understood this as being not unconditional, because it implies that the Emperor will still be able to be in control and the imperial house will continue to some (extremely vague) extent.

(The US correctly interpreted this in part because they were monitoring Japanese communications and knew that this was a major sticking point for the Japanese. To the Japanese high command, it was considered unacceptable to allow conditions that might destroy the imperial system, the kokutai, because they considered this synonymous with "Japan," in the same way that Americans today might consider the Constitution synonymous with the definition of "the United States.")

On August 11th, the US Secretary of State replied, saying, in essence, that they were not happy with that clause, and that surrender had to be unconditional — that the moment the surrender went into effect, Japan would be ruled by the Supreme Allied Authority, not the Emperor.

On August 14th, the Japanese conveyed an acceptance of the Potsdam terms completely, and on August 15th (after an attempted coup and other things), Hirohito gave a speech announcing as much.

In the end, the Allies agreed to let the Japanese maintain the Emperor as a more or less ceremonial figurehead, to maintain political/cultural stability, but this was not decided until the postwar period.

For a deep discussion of the back and forth that went on here, and the respective thinking and events on each side that led to these statements and replies, Hasegawa's Racing the Enemy is very good.