There is a strong argument to be made that it was not necessary to drop any atomic bombs on Japan to bring an end to the war. The Japanese leadership wanted a negotiated end to the war and was willing to make concessions to get it well before August 1945. Essentially it was the allies insistence on unconditional surrender in the Potsdam Declaration that was stopping the war from ending. So the allies really could have stopped the war at their convenience.

The question of whether dropping atomic bombs at all was necessary to get the Japanese to unconditionally surrender is more difficult to answer. This is because another pivotal thing happened between Hiroshima and Nagasaki: The USSR declared war and invaded in north mainland asia. The Japanese were overwhelmingly taken by surprise by this, previously hoping that Stalin would remain neutral, and possibly help negotiate a more beneficial peace for them. So this took one of their last desperate hopes away (along with a large slice of asian mainland).

In terms of the impact of the atomic bombs themselves, people often do not take into account that Japan was being heavily bombed already. Although Hiroshima was certainly a big shock, there is / was an argument to be made that lots of smaller bombs would have essentially the same effect as a few big ones on the people they were dropped on. While it did have a major sobering effect, the Hiroshima bomb did not by itself make the Japanese leadership decide to unconditionally surrender, and the Nagasaki one didn't really change their position at all. In fact, though Nagasaki confirmed the US had more than one nuclear weapon, the Japanese already had (false) intelligence that the US had 100 of them anyway.

The prospect of an invasion by the USSR combined with the knowledge the US had nuclear bombs was what really drove the Japanese to accept unconditional surrender. Which of those was more important is a matter of conjecture. But dropping the second atomic bomb on Nagasaki was probably not pivotal to the decision.

For a more lengthy, professional, heavily sourced discussion of this that makes the argument that Russian invasion was more pivotal to Japanese surrender than nuclear weapons you can look here: http://www.japanfocus.org/-Tsuyoshi-Hasegawa/2501

So this has been a debate amongst historians for a long time. The reason is that while a lot of people feel you can justify Hiroshima, it's a lot harder to justify Nagasaki only three days later. There is additionally a separate argument about whether the bombs really were what led to the Japanese surrender anyway.

On the surrender question, the argument is basically that the Soviet declaration of war, and invasion of Manchuria, were what led to the Japanese decision to surrender when they did. The reason here is that the Japanese were pinning their end-of-war hopes on a diplomatic intervention by a neutral USSR, and the sudden removal of that, coupled with the massive military forces of the USSR, made it clear that there was just no way out of it that didn't involve an intolerable amount of bloodshed. There are points in this argument's favor; the best exposition of it is Tsuyoshi Hasegawa's Racing the Enemy: Truman, Stalin, and the Surrender of Japan. Hasegawa thinks that Japan would have folded well before the American invasion, Operation Downfall, scheduled in November 1945, even if the bombs hadn't been used.

On the Nagasaki question, I've written at length about it here. The summary version is that it isn't clear whether Truman really intended the bombs to be dropped in such quick succession — the dropping of an atomic bomb was explicitly timed (first good weather date after August 3rd), but no tempo was given to how often they'd be dropped after that. The people on the ground seem to have just treated it as another weapon that you'd move out the door as soon as you could. Truman put a "stop" on atomic bombing on August 10th, when rumors came in that the Japanese were considering surrender, and also probably because he realized he was losing control of the situation. There are other theories, but this one — that it was not especially planned out — is the one which appeals to me. Michael Gordin's Five Days in August is one of the few books which discusses this question explicitly.