What were the reasons against dropping the atomic bombs on Japan?
At the time that the decision to drop the bomb was actually made, there was very little discussion of the issue. Winston Churchill, for example, said later that it was just assumed that the bomb would be used if it became available. There was no extensive discussion on any formal level.
There are plenty of people who stated after the fact that they had believed that the use of the bomb was not necessary. However, in most cases, it is not at all clear that they actually believed that in the days before the bomb was dropped. If they did in fact believe it, they did not make any strong arguments against using the bomb.
The major argument that people made in 1945 and the following years was that Japan was ready to surrender. They argue that the war was going so badly for the Japanese that they would inevitably have given up without the use of the bomb or an invasion.
In later years, other arguments have sprung up. More people have argued that it was immoral to use the bomb because it was in some way worse than destroying cities with conventional bombs. They have argued that the US used it for racist reasons. They have argued that the US used it only to scare the Soviet Union. These arguments, however, were not made at the time.
Thus, if anyone really disagreed with using the bomb in 1945, it was because they thought the Japanese would surrender even if the US did not use the bomb.
The moral dilemma was probably the most frequently cited reason against dropping an atomic weapon on Japan. It is fairly well countered by the idea that if the United States had not used the bomb, they would have had to invade the Japanese homeland, which would have resulted in a tremendous loss of life for the US military. Keeping in mind that Japan instigated the war by attacking Pearl Harbor makes it easier to drop the moral argument.
Another reason to choose not to use a nuclear weapon is the "Pandora's Box" argument. Once such a weapon has been used the first time, might it become easier for other countries to follow suit? It is true that at that time, only the United States possessed a nuclear weapon, but they knew full well that other countries would also develop them in time. It's probably surprising that, given the volatility of the world and the number of armed conflicts going on at any given time, another nuclear weapon (besides the two the US used in Japan) has not yet been used by somebody.