If you're talking about the conventional bombing that took place after Nagasaki (August 9) — basically, most serious bombing against Japan was halted after August 10th, because the Japanese sent a "conditional" surrender to the Americans. This was not what the Americans wanted (it still retained a condition regarding the status of the Emperor), and they rejected it. Because talks appeared "stalled" after that (in reality, there was an attempted coup and many other things going on in Japan), they agreed to conduct additional conventional bombing raids on August 14th, believing this would reinvigorate the surrender process (I am not sure there is any evidence that it had an effect). On August 14th, the Japanese issued its unconditional surrender, which was accepted by the US, and bombing stopped (except for those planes already in the air at the time it was issued).
An open question: would another atomic bomb have been used if they had continued to not accept unconditional surrender? (Another one would have been ready for use around August 17th.) It isn't clear, but Truman did indicate that he thought it might be necessary early in the day on August 14th. But this is somewhat speculative, as is the question of exactly how it would be used if so.
Hi there -- the question of why the US dropped a second atomic bomb on Japan, and whether it was the threat of more atomic bombs, or the Soviet invasion, or a combination, that triggered Japan's surrender has been asked here in a few ways. Not discouraging further comment, but you might want to start with this section of the FAQ on the atomic bombs. Specifically, these questions stand out:
What happened to the Japanese political/military landscape between August 6th, 1945 (the day that Little Boy was dropped on Hiroshima) and August 15th, 1945 (the day they surrendered). How did they come to the decision that surrender was the best option, and was there much disagreement? by /u/ScipioAsina, /u/restricteddata, and /u/t-o-k-u-m-e-i
There has been some controversy on the true effect of the atomic bombing of Japan. Was it the bomb, or the Soviet declaration of war that ended WWII? by /u/t-o-k-u-m-e-i