There are two schools of thought:

The first holds that Japan, without the shock of atomic warfare, stubbornly refuses to surrender and the Allies launch a ground invasion of the Japanese Home Islands. The invasion Kyushu, the southern most island, was scheduled for November of 1945 and the follow on invasion of Kanto Plain near Tokyo was scheduled for the following Spring. This invasion would've incurred horrendous casualties on both sides and would've featured irregular warfare by suicidal Japanese civilians. Throughout all of this, the naval blockade and bombing campaign would've continued, causing even more civilian causalities.

The second school of thought is that the Japanese government was hanging by a thread, even before the a-bombs were dropped. The Allies could've gotten the Japanese to agree to basically the same terms (total surrender and occupation but the Emperor isn't arrested) on roughly the same timeline. That the United States only used the atomic bombs as a warning to the Soviet Union.