The newly overhauled FAQs section has a number of excellent answers on this, divisive to say the least, subject.

The main controversy, from what I've seen, is the question of whether or not it was the Bombs or the Soviet Manchurian Offensive that prompted the Japanese surrender. Again, from what I've read and seen discussed on the subreddit, it was a case of both: The Soviet Offensive brought about a period of crisis and debate for Japanese leadership, who had hoped that the soviets would uphold the non-aggression pact, and potentially intercede for Japan. That hope dashed, there was still talk of a fight to the finish; the Navy had little clout, and the Army pretty much trumped the civilians, being prepared to accept (according to the History Hell To Pay) 20 million Japanese combatant and non-combatant casualties as a price for the defense of the home islands. It was the demonstration of the immense power of two atom bombs, and the American bluff that there was more where that came from, that finally brought the government and the emperor to the point of surrender. Even then, Japanese officers attempted to prevent the emperor from giving his famous radio address.

Had the bombs not been dropped, two likely scenarios would have played out: Invasion of Japan, or Blockade/Aerial & Naval Bombardment of Japan. In both cases, fighting in China between Nationalist, Communist and Red Army forces against the Japanese would have continued; Australian and Commonwealth fighting in the Indies and southwest Pacific likely would have continued; British, Indian and Commonwealth fighting against the Japanese in Malaya and ultimately Indochina would have continued; so the debate about American casualties often ignores the wider Allied losses had the war continued (and the resulting heavy Japanese losses). In the event of an American Invasion or Blockade/Bombardment, the Japanese Army planned reprisals against Allied POWs, and the Invasion itself would probably have been incredibly bloody and brutal, for both sides, if previous fighting in the Pacific had taught the Americans anything.

Instead, a Soviet Invasion of Manchuria and two Atom Bombs proved to be all that it took. Tragic and mortifying as it is that peace came at such a terrible price (a price that, in the case of the bombed cities, continued to be paid via fallout and radiation poisoning), the cost of peace could have been much higher, and even more terrible for both sides, had events not played out the way they did.

The question of what would have happened has been studied quite a bit at the time and since. The US Strategic Bombing Survey of 1946 concluded that:

Based on a detailed investigation of all the facts, and supported by the testimony of the surviving Japanese leaders involved, it is the Survey's opinion that certainly prior to 31 December 1945, and in all probability prior to 1 November 1945, Japan would have surrendered even if the atomic bombs had not been dropped, even if Russia had not entered the war, and even if no invasion had been planned or contemplated.

Some other later authors have come to the same conclusion (like Tsuyoshi Hasegawa). The basic argument is that the Japanese high command was losing its will to continue anyway, and that Japan was on the verge of mass famine because of the very successful naval blockades.

It is also worth noting that even at the time, even before the bombs were used, the idea of the Soviets declaring war on the Japanese was considered to be possibly war-ending in and of itself.

We of course cannot say what would or would not happened. But the idea that it was bombs or nothing, or even that an invasion would have been required, is not at all obvious, nor necessarily supported by sources today or at the time itself.